triangular number

How Much of My $1596 Will the Taxman Get?

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Today’s Puzzle is a Game Called Taxman:

Tax season in the United States began just a few days ago. Figuring taxes is no fun, but playing Taxman is.

You see 56 cards below. Imagine that they are 56 envelopes, each containing the dollar amount written on them.

You can take whatever envelope you’d like, but the taxman must be able to take at least one envelope when you do. What envelopes will the taxman take when it is his turn? He will take EVERY available envelope that has a number on it that is a factor of the envelope you took. So if you started off taking envelope 56, the taxman would take envelopes 1, 2, 4, 7, 8, 14, and 28. Thus the taxman would start off getting a whole lot more money than you while also limiting envelopes you can choose later. For example, you wouldn’t be able to pick any prime numbers after taking 56  because the 1 would be gone, and the taxman must be able to take one envelope with every choice you make. When you get to the point that there are no more envelopes that you can take, the taxman gets to take ALL the remaining envelopes. You want the taxman to get less money than you do, and hopefully as little money as possible. I’ve included the factors of each of the numbers at the top of each “envelope” to make the game easier for both you and the taxman. You can print “envelopes” from this excel file Taxman & 1537-1544.  You can have a friend be the taxman as you play or you can play both roles.

How I Played the Game:

I took envelopes as I asked myself these questions:

What is the largest prime number on the table? 53. I take 53, and the only envelope the taxman gets is 1.

What is the largest number that is a prime number squared? 49. I take 49, and the only envelope left for the taxman to take is 7.

What is the largest multiple of 7 that has only one other available factor? 35. I take 35, and the only envelope the taxman can still get is 5.

What is the largest multiple of 5 that has only one other available factor? and so forth.

Here is the order I took the envelopes. It is not the only possible order to use, but it was one in which there were only nine cards at the end of the game for the taxman to claim.

The winner of the game is the one with the most money at the end of the game.

How Do We Know Who Won the Game?

We could add up all the money I got and compare it with all the money the taxman got, but that wouldn’t be much fun. We could add up all the taxman money and subtract it from 1596 to find my total, but I prefer a different way: I remove all the taxman’s money from the table and try to mix envelopes to match mine. I start with my envelope with the most cash, 56 = 8 + 1 + 47, so I put the taxman’s envelopes with those numbers next to my 56.

Matching the “envelopes” up like this will take some time, but it feels like playing a game. It may have taken me as long to play this part of the game as it took me to play Taxman because sometimes I had to remove perfectly good sums in order to use all the taxman’s envelopes in a sum. Here is how I matched the envelopes:

I know that I won the game because I had at least one envelope without a sum next to it. I can determine by  how much I won the game by adding 35 + 33 + 46 + 39 + 27 + 38 + 20 + 30 + 42. That sum tells me I won the game by $310.

How much did the taxman get?
(1596-310)/2 = 643.

How much did I get to keep?
643 + 310 = 953.

A Factor Tree for 1596:

If you know that 40² = 1600, and 1600 – 4 = 1596, we might recognize that 1596 is the difference of two perfect squares: 40² – 2². Then we can also know that
1596 = (40 – 2)(40 + 2) = 38 × 42.

Let’s make a factor tree from the factor pair 38 × 42:

Since 1596 has several factors, it has several other possible factor trees.

Factors of 1596:

  • 1596 is a composite number.
  • Prime factorization: 1596 = 2 × 2 × 3 × 7 × 19, which can be written 1596 = 2² × 3 × 7 × 19.
  • 1596 has at least one exponent greater than 1 in its prime factorization so √1596 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1596 = (√4)(√399) = 2√399
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 × 2 = 24. Therefore 1596 has exactly 24 factors.
  • The factors of 1596 are outlined with their factor pair partners in the graphic below.

More about the Number 1596:

The reason that I was dividing up $1596 between me and the taxman is that 1596 is the 56th triangular number. It is the 56th triangular number because
(56)(57)/2 = 1596.

1596 is the difference of two squares in four different ways:

1596 is the sum of the first 15 Fibonacci numbers, which also means it is one number less than the 17th Fibonacci number.

 

 

1431 is a Triangular Number and a Hexagonal Number

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If you only look at a list of triangular numbers or a list of hexagonal numbers, you might miss the relationship that figurate numbers have with each other.

1378 is the 52nd triangular number, and you can use it to find the 53rd triangular number (1431), the 53rd square number, the 53rd pentagonal number, and so forth.

351 is the 26th triangular number, and you can use it to find the 27th triangular number, the 27th square number, the 27th pentagonal number, the 27th hexagonal number (1431), and so forth.

See the relationship in the graphic below:

Should you get excited that 1431 is BOTH a triangular number and a hexagonal number? Not really. It turns out that every hexagonal number is also a triangular number. (But not every triangular number is a hexagonal number.)

Here are a few more facts about the number 1431:

  • 1431 is a composite number.
  • Prime factorization: 1431 = 3 × 3 × 3 × 53, which can be written 1431 = 3³ × 53.
  • 1431 has at least one exponent greater than 1 in its prime factorization so √1431 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1431 = (√9)(√159) = 3√159.
  • The exponents in the prime factorization are 3 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1431 has exactly 8 factors.
  • The factors of 1431 are outlined with their factor pair partners in the graphic below.

1431 is the hypotenuse of a Pythagorean triple:
756-1215-1431 which is 27 times (28-45-53)

The Shape of 1326

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1326 is the 51st triangular number because 26 × 51 = 1326 and 2(26) is 1 more than 51.

Since 51 is an odd number, 1326 is also a hexagonal number. It is the 26th hexagonal number because 26 is half of 52 which is 1 more than 51.

Triangular numbers aren’t hard to visualize, but hexagonal numbers might be. Here are 1326 tiny dots showing that 1326 is a hexagonal number:

Here are some more facts about the number 1326:

  • 1326 is a composite number.
  • Prime factorization: 1326 = 2 × 3 × 13 × 17
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1326 has exactly 16 factors.
  • Factors of 1326: 1, 2, 3, 6, 13, 17, 26, 34, 39, 51, 78, 102, 221, 442, 663, 1326
  • Factor pairs: 1326 = 1 × 1326, 2 × 663, 3 × 442, 6 × 221, 13 × 102, 17 × 78, 26 × 51, or 34 × 39
  • 1326 has no square factors that allow its square root to be simplified. √1326 ≈ 36.41428

Here are some factor trees for 1326 that won’t topple over.

1326 is also the long side of four different Pythagorean triple triangles:
840-1026-1326 which is 6 times (140-171-221)
126-1320-1326 which is 6 times (21-220-221)
510-1224-1326 which is (5-12-13) times 102
624-1170-1326 which is (8-15-17) times 78

1275 is the 50th Triangular Number AND the 500th Number Whose Square Root Can Be Simplified

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I found two reasons to celebrate the number 1275: It is the 50th Triangular number, and it is the 500th number whose square root can be simplified.

First I’ll celebrate its square root by listing the 401st to the 500th numbers with simplifiable square roots. Having three or more simplifiable square roots in a row doesn’t happen that often, so I like to highlight them when it happens. 1274 and 1275 are highlighted because 1276 also has a square root that can be simplified:

If you’re wondering what are the first 400 numbers with simplifiable square roots, you can click on the graphics below that will give you 100 at a time:

1st 100 reducible square roots 2nd 100 reducible square roots Reducible Square Roots 516-765

Now to celebrate that 1275 is the 50th triangular number, I’ve arranged $12.75 in a triangle:

1275 can also be evenly divided by 5, and 25, in other words, by nickels and quarters!

Nickels won’t make a triangle but they can form a trapezoid. Here’s how I made this one: 1275 ÷ 5 = 255 which is 300 (the 24th triangular number) minus 45 (the 9th triangular number). Thus we can make $12.75 by arranging 255 nickels in a trapezoid with a top base of 10, a bottom base of 24, and a height of 15.

We can also use quarters to make a trapezoid. Here’s what I did: 1275 ÷ 25 = 51 which is 66 (the 11th triangular number) minus 15 (the 5th triangular number). Thus, $12.75 can be made by arranging 51 quarters in a trapezoid with a top base of 6, a bottom base of 11, and a height of 6.

Can you find any rectangular ways to arrange the coins to total $12.75?

Here’s a little more about the number 1275:

  • 1275 is a composite number.
  • Prime factorization: 1275 = 3 × 5 × 5 × 17, which can be written 1275 = 3 × 5² × 17
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1275 has exactly 12 factors.
  • Factors of 1275: 1, 3, 5, 15, 17, 25, 51, 75, 85, 255, 425, 1275
  • Factor pairs: 1275 = 1 × 1275, 3 × 425, 5 × 255, 15 × 85, 17 × 75, or 25 × 51,
  • Taking the factor pair with the largest square number factor, we get √1275 = (√25)(√51) = 5√51 ≈ 35.70714

1275 is the hypotenuse of SEVEN Pythagorean triples:
195-1260-1275 which is 15 times (13-84-85)
261-1248-1275 which is 3 times (87-416-425)
357-1224-1275 which is (7-24-25) times 51
540-1155-1275 which is 15 times (36-77-85)
600-1125-1275 which is (8-15-17) times 75
765-1020-1275 which is (3-4-5) times 255
891-912-1275 which is 3 times (297-304-425)

1225 is a Triangular Number, a Perfect Square, and . . .

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1 and 36 are both triangular numbers and square numbers. The next number that can make the same claim is 1225. Why is this so?

1225 is a triangular number that just happens to be the sum of two consecutive triangular numbers. Two consecutive triangular numbers can always be made into a perfect square. Here’s a gif that tries to illustrate these facts about 1225:

1225 Perfect Square and Triangular Number

GIFs like this at MakeaGif

1225 is the 49th triangular number because 49(50)/2 =1225

It is the 35th square number because 35² = 1225

That it is both a triangular number AND a square number is pretty remarkable. But guess what? 1225 is ALSO a hexagonal number.

1225 is the 25th hexagonal number because (25)(2·25 – 1)= 1225
In fact, 1225 is the smallest number greater than 1 that is a triangle, a square, AND a hexagon! (Yeah, 1 is also all three and a whole more, but does 1 dot REALLY look like a triangle, a square, a hexagon and everything else all at the same time?)
All of these facts are great reasons to get very excited about the number 1225. Here are some more facts about this number:
  • 1225 is a composite number.
  • Prime factorization: 1225 = 5 × 5 × 7 × 7, which can be written 1225 = 5²× 7²
  • The exponents in the prime factorization are 2 and 2. Adding one to each and multiplying we get (2 + 1)(2 + 1) = 3 × 3 = 9. Therefore 1225 has exactly 9 factors.
  • Factors of 1225: 1, 5, 7, 25, 35, 49, 175, 245, 1225
  • Factor pairs: 1225 = 1 × 1225, 5 × 245, 7 × 175, 25 × 49, or 35 × 35
  • 1225 is a perfect square. √1225 = 35
Not only is 1225 a triangular number that is the sum of two other triangular numbers, but
1225 is also a square that is the sum of two other squares!
28² + 21² = 35² = 1225

1225 is the hypotenuse of two Pythagorean triples:
735-980-1225 which is (3-4-5) times 245
343-1176-1225 calculated from 28² – 21², 2(28)(21), 28² + 21²,
but it is also (7-24-25) times 49

1225 looks like a square in some other bases:
It’s 441 in BASE 17,
169 in BASE 32,
144 in BASE 33,
121 in BASE 34, and
100 in BASE 35

OEIS.org reports that 1225 is the smallest number that can be written as the sum of four cubes three different ways. It looks like these are the three ways:
10³ + 6 ³ + 2³ + 1³ = 1225
9³ + 6³ + 6³ + 4³ = 1225
8³ + 7³ + 7³ + 3³ = 1225

1225 is, indeed, the smallest number that has all the special properties listed above!

903 Mystery Level

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903 can be written as the sum of consecutive numbers seven different ways:

  • 451 + 452 = 903; that’s 2 consecutive numbers.
  • 300 + 301 + 302 = 903; that’s 3 consecutive numbers.
  • 147 + 148 + 149 + 150 + 151 + 153 = 903; that’s 6 consecutive numbers.
  • 126 + 127 + 128 + 129 + 130 + 131 + 132 = 903; that’s 7 consecutive numbers.
  • 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69 + 70 + 71 = 903; that’s fourteen consecutive numbers.
  • 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 = 903; that’s 21 consecutive numbers.
  • 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 = 903; that’s 42 consecutive numbers.

That last way means that 903 is the 42nd triangular number. It happened because (42 × 43)/2 = 903.

Print the puzzles or type the solution on this excel file: 10-factors-897-904

  • 903 is a composite number.
  • Prime factorization: 903 = 3 × 7 × 43
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 903 has exactly 8 factors.
  • Factors of 903: 1, 3, 7, 21, 43, 129, 301, 903
  • Factor pairs: 903 = 1 × 903, 3 × 301, 7 × 129, or 21 × 43
  • 903 has no square factors that allow its square root to be simplified. √903 ≈ 30.0499584.

780 is the 39th Triangular Number

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  • 780 is a composite number.
  • Prime factorization: 780 = 2 x 2 x 3 x 5 x 13, which can be written 780 = (2^2) x 3 x 5 x 13
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 3 x 2 x 2 = 24. Therefore 780 has exactly 24 factors.
  • Factors of 780: 1, 2, 3, 4, 5, 6, 10, 12, 13, 15, 20, 26, 30, 39, 52, 60, 65, 78, 130, 156, 195, 260, 390, 780
  • Factor pairs: 780 = 1 x 780, 2 x 390, 3 x 260, 4 x 195, 5 x 156, 6 x 130, 10 x 78, 12 x 65, 13 x 60, 15 x 52, 20 x 39, or 26 x 30
  • Taking the factor pair with the largest square number factor, we get √780 = (√4)(√195) = 2√195 ≈ 27.92848.

780-factor-pairs

There are MANY ways to make factor trees for 780. Here are just three of them:

780 Factor Trees

OEIS.org informs us that (7 + 5)(8 + 5)(0 + 5) = 780.

780 is the sum of consecutive prime numbers two different ways:

  • 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 = 780 (10 consecutive primes).
  • 191 + 193 + 197 + 199 = 780 (4 of my favorite consecutive primes).

26 and 30 are both exactly 2 numbers away from their average 28, so 780 can be written as the difference of two squares:

780 = 26 x 30 = (28 – 2)(28 + 2) = 28² – 2² = 784 – 4.

(It can be written as the difference of two squares three other ways, but I won’t list them here.)

Because 5 and 13 are both factors of 780, it is the hypotenuse of FOUR Pythagorean triangles making each of these equations true:

  • 192² + 756² = 780²
  • 300² + 720² = 780²
  • 396² + 672² = 780²
  • 468² + 624² = 780²

780 can be written as the sum of three perfect squares two different ways:

  • 26² + 10² + 2² = 780
  • 22² + 14² + 10² = 780

780 BASE 10 is palindrome QQ BASE 29. (Q is 26 in BASE 10.) Note that 26(29) + 26(1) = 780.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 = 780, making 780 the 39th triangular number. Since 20 is in the exact middle of that list of numbers, 780 is also the 20th hexagonal number.

Triangular numbers are interesting, but are they good for anything? Here’s one good thing:

Count the Terms of Sums Squared

There are 26 letters in the English alphabet. 39 variables would use the alphabet exactly one and one half times, but that’s okay because we can use upper case letters the second time around:

(a + b + c + . . . + x + y+ z + A + B + C + . . . + K + L + M)² has exactly 780 terms because it has 39 single variables, and 780 is the 39th triangular number.

780 is the 39th triangular number because 39⋅40/2 = 780.

A couple of months ago I saw a fascinating image on twitter. It is a square made with 1001 dots representing the numbers from -500 to +500 with zero in the exact middle. The triangular numbers are represented by the “+” pattern seen here. If we added about 600 more dots to the square, -780 and +780 would also lie on that +.

//platform.twitter.com/widgets.js

Half of all triangular numbers are also hexagonal numbers. If there were enough dots, then -780 and +780 would also be among the dots forming the “-” pattern representing the hexagonal numbers. 780 is the 20th hexagonal number because 20(2⋅20-1) = 780.

 

What Kind of Shape is 703 in?

/ ivasallay / 3 Comments
  • 703 is a composite number.
  • Prime factorization: 703 = 19 x 37
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 703 has exactly 4 factors.
  • Factors of 703: 1, 19, 37, 703
  • Factor pairs: 703 = 1 x 703 or 19 x 37
  • 703 has no square factors that allow its square root to be simplified. √703 ≈ 26.514147.

703 is a fascinating number:

Kaprekar number 703

As shown in that graphic, since the square of 703 can be broken apart and added back together to become itself again, 703 is the 7th Kaprekar number. Thank you OEIS.org for that fun fact.

What kind of shape is 703 in?

703 is the 37th triangular number:

703 is 37th triangular number

703 is a triangular number because 37(38)/2 = 703. If we move part of the triangle, those blocks can be neatly rearranged into this rectangle:

703 rectangle

About half of all triangular numbers are also hexagonal numbers. 703 is the 19th hexagonal number because (19⋅2)(19⋅2 – 1)/2 = 703.

In the past I found counting the dots in graphics of larger hexagonal numbers to be a nightmare, but yesterday I saw a small graphic for hexagonal number 28 that used different colors for each expansion. I challenged myself to make a graphic showing that 703 is a hexagonal number. I made it in excel using o’s in different colors. When I finished, I told excel to replace the o’s with •’s. Excel informed me that it made 703 replacements.  Then I adjusted the size of the cells to make the hexagon smaller. I also attempted to make the outer-most hexagon look as symmetrical as possible.

703 is the 19th Hexagonal Number

Hexagonal numbers are not symmetrical the way hexagonal snowflakes are. I thought it would be cool to make 703 into a snowflake, but I didn’t succeed because 703 is just too big and it lacks symmetry. Nevertheless, I must share these directions to make snowflakes that are indeed 6-sided. (Using paper dinner napkins instead of regular paper makes folding and cutting them much simpler.)

703 is a palindrome in three different bases:

  • 383 base 14; note that 3(196) + 8(14) + 3(1) = 703
  • 111 base 26; note that 1(676) + 1(26) + 1(1) =703
  • JJ base 36; note that J is equivalent to 19 in base 10 and 19(36) + 19(1) = 703

Since 37 is one of its factors, 703 is the hypotenuse of the Pythagorean triple we see in this triangle:

703 Hypotenuse

If you divide each number in the triple by 19, you’ll get the primitive Pythagorean triple 12-35-37.

703 is also a leg in some Pythagorean triple triangles:

  • 504-703-865 (a primitive generated from 2(28)(9), 28² – 9², 28² + 9²) Note that 28 + 9 = 37 and 28 – 9 = 19. Both 37 and 19 are prime factors of 703.
  • 703-6660-6697
  • 703-12996-13015
  • 703-247104-247105 (another primitive)

703 Legs

 

 

630 Factor Trees and Level 2

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Today’s Puzzle:

630 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-28

Factors for Today’s Puzzle:

630 Factors

Factor Trees for 630:

630 is the 7th number with exactly 24 factors. So far, the seven numbers counting numbers with 24 factors are 360, 420, 480, 504, 540, 600, and 630. No counting number less than 630 has more than 24 factors.

Two of those seven numbers make up the Pythagorean triple 378-504-630. Which factor of 630 is the greatest common factor of those three numbers in the triple?

Here are a few of the MANY possible factor trees for 630.

630 Factor Trees

Factors of 630:

  • 630 is a composite number.
  • Prime factorization: 630 = 2 x 3 x 3 x 5 x 7, which can be written 630 = 2 x (3^2) x 5 x 7
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 x 3 x 2 x 2 = 24. Therefore 630 has exactly 24 factors.
  • Factors of 630: 1, 2, 3, 5, 6, 7, 9, 10, 14, 15, 18, 21, 30, 35, 42, 45, 63, 70, 90, 105, 126, 210, 315, 630
  • Factor pairs: 630 = 1 x 630, 2 x 315, 3 x 210, 5 x 126, 6 x 105, 7 x 90, 9 x 70, 10 x 63, 14 x 45, 15 x 42, 18 x 35, or 21 x 30
  • Taking the factor pair with the largest square number factor, we get √630 = (√9)(√70) = 3√70 ≈ 25.09980.

Sum-Difference Puzzle:

630 has twelve factor pairs. One of the factor pairs adds up to 53, and a different one subtracts to 53. If you can identify those factor pairs, then you can solve this puzzle!

More about the Number 630:

630 is the sum of the six prime numbers from 97 to 113.

630 is the 35th triangular number because (35 x 36)/2 = 630. It is also the 18th hexagonal number because 18(2 x 18 – 1) = 630.

630 is a triangular number that is a multiple of other triangular numbers in more ways than you probably want to know:

  • 630 is three times the 20th triangular number, 210, because 3(20 x 21)/2 = 630.
  • 630 is 6 times the 14th triangular number, 105, because 6(14 x 15)/2 = 630.
  • 630 is 14 times the 9th triangular number, 45, because 14(9 x 10)/2 = 630.
  • 630 is 30 times the 6th triangular number, 21, because 30(6 x 7)/2 = 630.
  • 630 is 42 times the 5th triangular number, 15, because 42(5 x 6)/2 = 630.
  • 630 is 63 times the 4th triangular number, 10, because 63(4 x 5)/2 = 630.
  • 630 is 105 times the 3rd triangular number, 6, because 105(3 x 4)/2 = 630.
  • 630 is 210 times the 2nd triangular number, 3, because 210(2 x 3)/2 = 630.
  • and finally, 630 is 630 times the 1st triangular number, 1, because 630(1 x 2)/2 = 630.

571 Family Time

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For the last two weeks we’ve been spending time with our grandchildren in Salt Lake City, Utah; Portland, Oregon; Houston, Texas; and Hartford, Connecticut. We also spent time with my brother’s daughter and her two children. AND we celebrated my sister’s 70th birthday party and reconnected with all of her children in Portland. We have been busy, and I am tired, but happy.

We’ve had a wonderful time with all of them, and it was so painful to say goodbye. We and a few other people took pictures except we didn’t get any pictures of the grandkids that live within 10 miles of our home. What’s with that? I’m as guilty as anyone else who takes people and things for granted.

Here’s a picture of me holding my newest grandson. My daughter is in the background holding one of her nieces.

 

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Since 171 + 190 + 210 = 571, it is the sum of the 18th, 19th, and 20th triangular numbers. That makes 571 the 20th centered triangular number.

  • 571 is a prime number and a twin prime with 569.
  • Prime factorization: 571 is prime.
  • The exponent of prime number 571 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 571 has exactly 2 factors.
  • Factors of 571: 1, 571
  • Factor pairs: 571 = 1 x 571
  • 571 has no square factors that allow its square root to be simplified. √571 ≈ 23.8956

How do we know that 571 is a prime number? If 571 were not a prime number, then it would be divisible by at least one prime number less than or equal to √571 ≈ 23.9. Since 571 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 571 is a prime number.