## A Multiplication Based Logic Puzzle

### 780 is the 39th Triangular Number

• 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.

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

Stetson.edu 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:

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 +.

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?

• 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:

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 Stetson.edu for that fun fact.

What kind of shape is 703 in?

703 is the 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:

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.

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:

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)

### 630 Factor Trees and Level 2

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 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

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

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• 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.

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### 571 Family Time

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 and cannot be factored.
• 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.

### 526 is a Centered Pentagonal Number

I am excited that Kerrydunton included the Find the Factors puzzles on a list of great maths starters.

1 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 + 50 + 55 + 60 + 65 + 70 = 526. Thus, 526 is the 15th centered pentagonal number.

The above equation is the same as saying that 526 is one more than five times the 14th triangular number.

The last digit of a centered pentagonal number is always 1 or 6. Triangular numbers greater than 3 are always composite numbers, but centered pentagonal numbers ending in 1 might be prime numbers. For example 31, 181, 331, 601, 1051, 1381, and 3331 are centered pentagonal numbers and prime numbers.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-15

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• 526 is a composite number.
• Prime factorization: 526 = 2 x 263
• 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 526 has exactly 4 factors.
• Factors of 526: 1, 2, 263, 526
• Factor pairs: 526 = 1 x 526 or 2 x 263
• 526 has no square factors that allow its square root to be simplified. √526 ≈ 22.93468988

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### 351 is a Triangular Number

351 is the 26th triangular number. The only triangular numbers that are NOT composite numbers are 1 and 3. All other triangular numbers are composite numbers because they can be written as the product of two consecutive numbers divided by 2. For example, 351 can be written like this:

• 1 + 2 + 3 + . . . + 24 + 25 + 26 = (26 x 27)/2 = 351

Also, 351 is divisible by 3 because 1, 3, 5 are three consecutive odd numbers. Since 3 is the middle number in that list, 351 is also divisible by 9. Scroll down past the puzzle to see the rest of 351’s factors.

Print the puzzles or type the factors on this excel file: 12 Factors 2015-01-12

• 351 is a composite number.
• Prime factorization: 351 = 3 x 3 x 3 x 13, which can be written (3^3) x 13
• The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 351 has exactly 8 factors.
• Factors of 351: 1, 3, 9, 13, 27, 39, 117, 351
• Factor pairs: 351 = 1 x 351, 3 x 117, 9 x 39, or 13 x 27
• Taking the factor pair with the largest square number factor, we get √351 = (√9)(√39) = 3√39 ≈ 18.735

### 210 and Level 4

• 210 is a composite number and a triangular number.
• Prime factorization: 210 = 2 x 3 x 5 x 7, so 210 is the product of the first four prime numbers!
• 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 x 2 x 2 x 2 = 16. Therefore 210 has 16 factors.
• Factors of 210: 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210
• Factor pairs: 210 = 1 x 210, 2 x 105, 3 x 70, 5 x 42, 6 x 35, 7 x 30, 10 x 21, or 14 x 15
• Since 210 has no square factors, √210 cannot be simplified. √210 ≈ 14.491

Because 20 + 1 = 21 and (20/2) x 21 = 210, we know that 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 = 210, thus it is the 20th triangular number.

Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-08-18