Mathematics

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)

1274 Imagining With Sain Smart Jr.’s Tetris Puzzle

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My granddaughter recently received a Sain Smart Jr. Tetris Puzzle for her birthday. I got to watch as she and her younger sister had lots of fun creating pictures using the colorful pieces. Here are some of their creations:

 

As they played they experimented and learned what they liked and what they didn’t like. In the process, they learned some mathematics and may not have even realized it.

Yeah, they could also make all the Tetris puzzle pieces fit in the frame. It seems to help if you save some of the five wood grained pieces for last.

From what I could tell, this was a very fun and educational toy.

Now here I’ll share some information about the number 1274:

  • 1274 is a composite number.
  • Prime factorization: 1274 = 2 × 7 × 7 × 13, which can be written 1274 = 2 × 7² × 13
  • 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 1274 has exactly 12 factors.
  • Factors of 1274: 1, 2, 7, 13, 14, 26, 49, 91, 98, 182, 637, 1274
  • Factor pairs: 1274 = 1 × 1274, 2 × 637, 7 × 182, 13 × 98, 14 × 91, or 26 × 49,
  • Taking the factor pair with the largest square number factor, we get √1274 = (√49)(√26) = 7√26 ≈ 35.69314

35² + 7² = 1274

1274 is the hypotenuse of a Pythagorean triple:
490-1176-1274 which is (5-12-13) times 98.
It is also 2(35)( 7), 35² – 7², 35² + 7²

1266 is a Centered Pentagonal Number

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1266 is the 23rd centered pentagonal number because 5(22)(23)/2 + 1 = 1266. The graphic below shows 1266 tiny dots arranged into a pentagonal shape and that 1266 is one more than five times the 22nd triangular number.

Here are some more facts about the number 1266:

  • 1266 is a composite number.
  • Prime factorization: 1266 = 2 × 3 × 211
  • 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 1266 has exactly 8 factors.
  • Factors of 1266: 1, 2, 3, 6, 211, 422, 633, 1266
  • Factor pairs: 1266 = 1 × 1266, 2 × 633, 3 × 422, or 6 × 211
  • 1266 has no square factors that allow its square root to be simplified. √1266 ≈ 35.58089

1266 is also the sum of ten consecutive prime numbers:
103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 = 1266

 

1261 Can You Make a Star out of a Hexagon?

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Can you make a star out of a hexagon? If you have 1261 tiny squares arranged as a centered hexagon, you can rearrange those 1261 tiny squares into a six-pointed star as illustrated below!

37 was the last centered hexagonal number that was also a star number.

Here are some more facts about the number 1261:

  • 1261 is a composite number.
  • Prime factorization: 1261 = 13 × 97
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1261 has exactly 4 factors.
  • Factors of 1261: 1, 13, 97, 1261
  • Factor pairs: 1261 = 1 × 1261 or 13 × 97
  • 1261 has no square factors that allow its square root to be simplified. √1261 ≈ 35.51056

1261 is the sum of two squares two different ways:
30² + 19² = 1261
35² + 6² = 1261

1261 is the hypotenuse of FOUR Pythagorean triples:
420-1189-1261 calculated from 2(35)(6), 35² – 6², 35² + 6²
485-1164-1261 which is (5-12-13) times 97
539-1140-1261 calculated from 30² – 19², 2(30)(19), 30² + 19²
845-936-1261 which is 13 times (65-72-97)

Haunted Forest with 1260 Factor Trees

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1260 is the smallest number with 36 factors. That’s a new record. (32 was the old record and was held by both 840 and 1080.)

Often when a number has a lot of factors, we will visit a forest of its factor trees. 1260 certainly deserves such a forest. Since it is just before Halloween, It happens to be a haunted forest. Do you dare to go into such a forest? These three trees are scary enough for me! However, there are MANY more factor trees in that haunted forest! Perhaps if you are brave, you can find some of those factor trees in the haunted forest yourself.

Here’s more about the number 1260:

  • 1260 is a composite number.
  • Prime factorization: 1260 = 2 × 2 × 3 × 3 × 5 × 7, which can be written 1260 = 2² × 3² × 5 × 7.
  • The exponents in the prime factorization are 2, 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) (1 + 1) = 3 × 3 × 2 × 2  = 36. Therefore 1260 has exactly 36 factors.
  • Factors of 1260: 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 21, 28, 30, 35, 36, 42, 45, 60, 63, 70, 84, 90, 105, 126, 140, 180, 210, 252, 315, 420, 630, 1260
  • Factor pairs: 1260 = 1 × 1260, 2 × 630, 3 × 420, 4 × 315, 5 × 252, 6 × 210, 7 × 180, 9 × 140, 10 × 126, 12 × 105, 14 × 90, 15 × 84, 18 × 70, 20 × 63, 21 × 60, 28 × 45, 30 × 42 or 35 × 36
  • Taking the factor pair with the largest square number factor, we get √1260 = (√36)(√35) = 6√35 ≈ 35.49648

21 × 60 = 1260 The same digits are used on both sides of that equation and that makes 1260 the 19th Friedman number.

1260 is also the sum of the interior angles of a nine-sided polygon. Convex or concave, that is the sum. The concave nonagon below is a good illustration of that fact:

1260 is also the hypotenuse of a Pythagorean triple:
756-1008-1260 which is (3-4-5) times 252

1252 What Do You Notice?

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Remembering that 1250 = 25² + 25² from just a couple of days ago, I was struck when I noticed that 1252 = 24² + 26². I wondered if it was part of a pattern, so I made this chart. What do you think?

When I thought about it more, I realized that perhaps it isn’t so remarkable. After all,
(n – 1)² + (n + 1)² = (n² – 2n +1) + (n² + 2n +1) = n² + n² – 2n + 2n + 1 + 1  = 2n² + 2
Nevertheless, I still like the pattern.

Here are some more facts about the number 1252:

  • 1252 is a composite number.
  • Prime factorization: 1252 = 2 × 2 × 313, which can be written 1252 = 2² × 313
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 1252 has exactly 6 factors.
  • Factors of 1252: 1, 2, 4, 313, 626, 1252
  • Factor pairs: 1252 = 1 × 1252, 2 × 626, or 4 × 313
  • Taking the factor pair with the largest square number factor, we get √1252 = (√4)(√313) = 2√313 ≈ 35.38361

Because 1252 = 26² + 24², it is the hypotenuse of a Pythagorean triple:
100-1248-1252 calculated from 26² – 24², 2(26)(24), 26² + 24²

100-1248-1252 is also 4 times (25-312-313)

 

1247 Is a Pentagonal Number

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Two factors of 1247 make it the 29th pentagonal number. Here’s why:

29(3·29-1)/2 = 29(86)/2 = 29(43) = 1247

Here is an illustration of this pentagonal number featuring a different, but equivalent, formula.  Seeing the pentagonal numbers less than 1247 in the illustration won’t be difficult either.

Here are some more facts about the number 1247:

  • 1247 is a composite number.
  • Prime factorization: 1247 = 29 × 43
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1247 has exactly 4 factors.
  • Factors of 1247: 1, 29, 43, 1247
  • Factor pairs: 1247 = 1 × 1247 or 29 × 43
  • 1247 has no square factors that allow its square root to be simplified. √1247 ≈ 35.31289

1247 is the sum of consecutive prime numbers two different ways:
It is the sum of the twenty-three prime numbers from 11 to 103.
It is also the sum of seven consecutive primes:
163 + 167 + 173 + 179 + 181 + 191 + 193 = 1247

1247 is the hypotenuse of a Pythagorean triple:
860-903-1247 which is (20-21-29) times 43

 

1242 is a Decagonal Number

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If you had 1242 tiny little squares you could arrange them into a decagon, just as I did for the graphic below.

18 is a factor of 1242. Since 18(4·18-3) = 18(69) = 1242, it is the 18th decagonal number.

Here are a few more facts about the number 1242:

  • 1242 is a composite number.
  • Prime factorization: 1242 = 2 × 3 × 3 × 3 × 23, which can be written 1242 = 2 × 3³ × 23
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 × 2 × 2 = 16. Therefore 1242 has exactly 16 factors.
  • Factors of 1242: 1, 2, 3, 6, 9, 18, 23, 27, 46, 54, 69, 138, 207, 414, 621, 1242
  • Factor pairs: 1242 = 1 × 1242, 2 × 621, 3 × 414, 6 × 207, 9 × 138, 18 × 69, 23 × 54, or 27 × 46
  • Taking the factor pair with the largest square number factor, we get √1242 = (√9)(√138) = 3√138 ≈ 35.24202

1242 is the sum of consecutive prime numbers two different ways:
It is the sum of the eighteen prime numbers from 31 to 107, and
it is also the sum of the sixteen prime numbers from 43 to 109.

1240 is a Square Pyramidal Number

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1240 is the 15th square pyramidal number because
1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10² + 11² + 12² + 13² + 14² + 15² = 1240

We can know that 1240 is the 15th square pyramidal number because
15(15 + 1)(2·15 + 1)/6
= 15(16)(31)/6
= (5)(8)(31)
= (40)(31)
= 1240

Here are some more facts about the number 1240:

  • 1240 is a composite number.
  • Prime factorization: 1240 = 2 × 2 × 2 × 5 × 31, which can be written 1240 = 2³ × 5 × 31
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 1240 has exactly 16 factors.
  • Factors of 1240: 1, 2, 4, 5, 8, 10, 20, 31, 40, 62, 124, 155, 248, 310, 620, 1240
  • Factor pairs: 1240 = 1 × 1240, 2 × 620, 4 × 310, 5 × 248, 8 × 155, 10 × 124, 20 × 62, or 31 × 40
  • Taking the factor pair with the largest square number factor, we get √1240 = (√4)(√310) = 2√310 ≈ 35.21363


1240 is the hypotenuse of a Pythagorean triple:
744-992-1240 which is (3-4-5) times 248

1239 Addition and Subtraction Families

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Perhaps in the early years of your education, you were introduced to addition and subtraction families. For example, you might have made a little house out of these four addition and subtraction facts:

3 + 9 = 12
9 + 3 = 12
12 – 9 = 3
12 – 3 = 9

You should have been told then, but weren’t, that there are other members of this addition and subtraction family:

12 = 3 + 9
12 = 9 + 3
3 = 12 – 9
9 = 12 – 3

In fact, those second four addition and subtraction facts may have seemed very strange-looking even years later.

Eventually, you should have been introduced to the whole family of facts involving addition and subtraction and those numbers, but most likely that never happened. Here, the most familiar part of the family can be seen in the dark green part of the house, but the entire rest of the family can also be seen throughout the rest of the house. Some are in the basement and some in the wings of the house, but they all very much belong in this addition and subtraction family home. And it isn’t too difficult to see where every member of the family came from:

If you knew all the members of that family, it would be more natural to accept the members of a family made with variables or numbers mixed with variables:

Instead, many students get very confused when they become teenagers and are introduced to families in which some family members are numbers and some are letters. Likewise, some family members are positive and some are negative.

Going from “a + b = c” to “c – a = b” becomes confusing instead of natural. Required steps involve adding and subtracting the same value from both sides of the equation instead of recalling prior knowledge known since first grade.

I think middle school students might benefit from building an addition and subtraction family house.

Now I would like to share some facts about the number 1239:

  • 1239 is a composite number.
  • Prime factorization: 1239 = 3 × 7 × 59
  • 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 1239 has exactly 8 factors.
  • Factors of 1239: 1, 3, 7, 21, 59, 177, 413, 1239
  • Factor pairs: 1239 = 1 × 1239, 3 × 413, 7 × 177, or 21 × 59
  • 1239 has no square factors that allow its square root to be simplified. √1239 ≈ 35.19943

1239 is divisible by 3 because it is made with three consecutive numbers and a multiple of 3. In this case it isn’t necessary to add the numbers up to see that.