Look at this chart of the sum of the factors for the numbers 1 – 25:

If we made the chart infinitely long using every counting number as n, there are certain numbers like 2, 5, 52, 88, and 96 that will NEVER appear in either column C or column D. Those numbers are called untouchable numbers, and 516 is one of them. Even though there are relatively few untouchable numbers, Paul Erdős proved that there are infinitely many of them.

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-08

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• 516 is a composite number.
• Prime factorization: 516 = 2 x 2 x 3 x 43, which can be written 516 = (2^2) x 3 x 43
• The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 516 has exactly 12 factors.
• Factors of 516: 1, 2, 3, 4, 6, 12, 43, 86, 129, 172, 258, 516
• Factor pairs: 516 = 1 x 516, 2 x 258, 3 x 172, 4 x 129, 6 x 86, or 12 x 43
• Taking the factor pair with the largest square number factor, we get √516 = (√4)(√129) = 2√129 ≈ 22.715633382

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515 = 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 which are nine consecutive prime numbers.

515 is the hypotenuse of the Pythagorean triple 309-412-515. What is the greatest common factor of those three numbers?

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-01

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• 515 is a composite number.
• Prime factorization: 515 = 5 x 103
• 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 515 has exactly 4 factors.
• Factors of 515: 1, 5, 103, 515
• Factor pairs: 515 = 1 x 515 or 5 x 103
• 515 has no square factors that allow its square root to be simplified. √515 ≈ 22.6936114

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A rhyme to help kids remember the even digits:

“0-2-4-6-8 being even is just great!”

That rhyme is often followed by another one to help kids remember the odd digits:

1-3-5-7-9 being odd is just fine!

Here’s a fun fact about the number 514:

514^3 = 135796744. OEIS.org tells us that 514 is the smallest number whose cube starts with all five of those odd digits in consecutive order.

514 is also the hypotenuse of the Pythagorean triple 64-510-514. What is the greatest common factor of those three numbers?

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-01

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• 514 is a composite number.
• Prime factorization: 514 = 2 x 257
• 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 514 has exactly 4 factors.
• Factors of 514: 1, 2, 257, 514
• Factor pairs: 514 = 1 x 514 or 2 x 257
• 514 has no square factors that allow its square root to be simplified. √514 ≈ 22.671568

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510 is the sum of consecutive prime numbers three different ways.

1. 510 = 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79
2. 510 = 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71
3. 510 = 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67

510 is the hypotenuse of four Pythagorean triples, none of which are primitive:

1. 78-504-510
2. 216-462-510
3. 240-450-510
4. 306-408-510

Two of those triples have the same greatest common factor. Can you identify which ones they are?

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-01

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• 510 is a composite number.
• Prime factorization: 510 = 2 x 3 x 5 x 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 x 2 x 2 x 2 = 16. Therefore 510 has exactly 16 factors.
• Factors of 510: 1, 2, 3, 5, 6, 10, 15, 17, 30, 34, 51, 85, 102, 170, 255, 510
• Factor pairs: 510 = 1 x 510, 2 x 255, 3 x 170, 5 x 102, 6 x 85, 10 x 51, 15 x 34, or 17 x 30
• 510 has no square factors that allow its square root to be simplified. √510 ≈ 22.58317958

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509 = 22^2 + 5^2.

509 is the hypotenuse of the primitive Pythagorean triple, 220-459-509.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-01

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• 509 is a prime number.
• Prime factorization: 509 is prime.
• The exponent of prime number 509 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 509 has exactly 2 factors.
• Factors of 509: 1, 509
• Factor pairs: 509 = 1 x 509
• 509 has no square factors that allow its square root to be simplified. √509 ≈ 22.561028

How do we know that 509 is a prime number? If 509 were not a prime number, then it would be divisible by at least one prime number less than or equal to √509 ≈ 22.6. Since 509 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 509 is a prime number.

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507 cannot be evenly divided by 4 or 9, but to simplify its square root, I would still make a little cake:

√507 = (√169)(√3) = 13√3

If I didn’t recognize that 169 is a perfect square, I would apply some prime number divisibility tricks in numerical order on 169:

• 2: 169 is not even so it’s not divisible by 2.
• 3: 1 + 6 + 9 = 16 which is not a multiple of 3 so 169 is not divisible by 3.
• 5: The last digit is not 0 or 5, so 169 cannot be evenly divided by 5.
• 7: The difference between 16 and 9 x 2 is 2 which is not a divisible by 7, so 169 is not divisible by 7.
• 11: 1 – 6 + 9 = 4 which is not a multiple of 11, so 169 is not divisible by 11.
• 13: To check if 169 is divisible by 13, let’s make a second layer to our cake:

Taking the square root of everything on the outside of the cake and multiplying it all together we get, √507 = (√3)(√13)(√13) = 13√3.

Here’s today’s puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-25

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• 507 is a composite number.
• Prime factorization: 507 = 3 x 13 x 13, which can be written 507 = 3 x (13^2)
• The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 507 has exactly 6 factors.
• Factors of 503: 1, 3, 13, 39, 169, 507
• Factor pairs: 507 = 1 x 507, 3 x 169, or 13 x 39
• Taking the factor pair with the largest square number factor, we get √507 = (√169)(√3) = 13√3 ≈ 22.51666

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