# 508 A Mathematical and Biblical Truth: 2 > 1

Two are better than one; because they have a good reward for their labour.

For if they fall, the one will lift up his fellow: but woe to him that is alone when he falleth; for he hath not another to help him up.

Ecclesiastes 4: 9 – 10

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508 is the sum of some consecutive prime numbers, and at least one of those prime numbers is in its prime factorization. Can you figure out what those consecutive primes are?

• 508 is a composite number.
• Prime factorization: 508 = 2 x 2 x 127, which can be written 508 = (2^2) x 127
• 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 508 has exactly 6 factors.
• Factors of 508: 1, 2, 4, 127, 254, 508
• Factor pairs: 508 = 1 x 508, 2 x 254, or 4 x 127
• Taking the factor pair with the largest square number factor, we get √508 = (√4)(√127) = 2√127 ≈ 22.53 88 55 33 9

# 507 and Level 6

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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# 506 and Level 5

506 is divisible by 11 because 5 + 6 – 0 = 11, and 11 obviously is divisible by 11.

506 is the 11th square pyramidal number because it is the sum of the first eleven square numbers.

Thus 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 + 100 + 121 = 506.

That was predictable because 506 = (11 x 12 x 23)/6 and 12 = 11 + 1 and 23 = 2(11) + 1.

Since 506 = 22 x 23, it is the sum of the first 22 even numbers which also happens to be exactly two times the 22nd triangular number, 253.

Now here’s a Level 5 puzzle for you to try:

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

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• 506 is a composite number.
• Prime factorization: 506 = 2 x 11 x 23
• 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 x 2 x 2 = 8. Therefore 506 has exactly 8 factors.
• Factors of 506: 1, 2, 11, 22, 23, 46, 253, 506
• Factor pairs: 506 = 1 x 506, 2 x 253, 11 x 46, or 22 x 33
• 506 has no square factors that allow its square root to be simplified. √506 ≈ 22.49444

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# 505 and Level 4

I learned the coolest visual fact about the number 505 from stetson.edu:

505 = 10C5 + 10C0 + 10C5.

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

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• 505 is a composite number.
• Prime factorization: 505 = 5 x 101
• 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 505 has exactly 4 factors.
• Factors of 505: 1, 5, 101, 505
• Factor pairs: 505 = 1 x 505 or 5 x 101
• 505 has no square factors that allow its square root to be simplified. √505 ≈ 22.47220505

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# 504 and Level 3

What do the numbers 360, 420, 480, and 504 have in common? The first three numbers are all multiples of 60, but 504 isn’t. 504 is the smallest number that isn’t divisible by 60 that has something special. Can you figure out what?

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

The first few digits of √504 is easy to memorize so I’ve made a little cake to help find it:

Just take the square root of everything on the outside of the cake and multiply them together: √504 = (√4)(√9)(√14) = (2 x 3)(√14) = 6√14 ≈ 22.449944

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• 504 is a composite number.
• Prime factorization: 504 = 2 x 2 x 2 x 3 x 3 x 7, which can be written 504 = (2^3) x (3^2) x 7
• The exponents in the prime factorization are 3, 2 and 1. Adding one to each and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 x 3 x 2 = 24. Therefore 504 has exactly 24 factors.
• Factors of 504: 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 18, 21, 24, 28, 36, 42, 56, 63, 72, 84, 126, 168, 252, 504
• Factor pairs: 504 = 1 x 504, 2 x 252, 3 x 168, 4 x 126, 6 x 84, 7 x 72, 8 x 63, 9 x 56, 12 x 42, 14 x 36, 18 x 28 or 21 x 24
• Taking the factor pair with the largest square number factor, we get √504 = (√36)(√14) = 6√14 ≈ 22.449944

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# 503 and Level 2

503 = (2^3) + (3^3) + (5^3) + (7^3) which is the sum of the cubes of the first four prime numbers. 503 is the smallest prime number that is the sum of consecutive cubes of prime numbers.

503 is also the sum of three consecutive prime numbers: 163, 167, and 173.

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

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• 503 is a prime number.
• Prime factorization: 503 is prime.
• The exponent of prime number 503 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 503 has exactly 2 factors.
• Factors of 503: 1, 503
• Factor pairs: 503 = 1 x 503
• 503 has no square factors that allow its square root to be simplified. √503 ≈ 22.42766

How do we know that 503 is a prime number? If 503 were not a prime number, then it would be divisible by at least one prime number less than or equal to √503 ≈ 22.4. Since 503 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 503 is a prime number.

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# 502 and Level 1

The prime factorization of 502 is 2 x 251. How many integers less than or equal to 502 do not have either of those numbers in their prime factorizations?

There is actually a function that counts how many. It is called the totient function or Euler’s totient function and looks like φ(502).

502 is the first integer that has both 2 and 251 in its prime factorization so finding φ(502) will be easy: first eliminate the 251 integers less than or equal to 502 that are divisible by 2. Then eliminate 251 because it is the only remaining number that is divisible by 502’s other prime factor. Thus φ(502) = 502 – 251 – 1 = 250.

Notice that 502 and 250 use the same digits. I learned this fact about the number 502 and φ(502) by reading Stetson.edu.

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

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• 502 is a composite number.
• Prime factorization: 502 = 2 x 251
• 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 502 has exactly 4 factors.
• Factors of 502: 1, 2, 251, 502
• Factor pairs: 502 = 1 x 502 or 2 x 251
• 502 has no square factors that allow its square root to be simplified. √502 ≈ 22.4053565

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# 501 and Level 6

2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 = 501. That was the first 18 prime numbers.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-18

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• 501 is a composite number.
• Prime factorization: 501 = 3 x 167
• 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 501 has exactly 4 factors.
• Factors of 501: 1, 3, 167, 501
• Factor pairs: 501 = 1 x 501 or 3 x 167
• 501 has no square factors that allow its square root to be simplified. √501 ≈ 22.383029

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# 500 Pick Your Pony! Who’ll Win This Number of Factors Horse Race?

Today I factor the number 500. How many factors does it have? Each number between 401 and 500 has at least 2 factors, but no more than 24 factors.

What if we had a horse race between the number of factors? Click on the graphic below to see a gif of the numbers racing against each other. Before you click, pick your pony. Will 2, 4, 6, 8, 9, 10, 12, 14, 16, 18, 20, or 24 be the number of factors of more integers between 401 and 500 than any other number? Click on the graphic to find out!

Did you see the lead change a couple of times? How did your pony do? Which pony will you choose in the 501 to 600 race?

Remarkably, only 37 of these one hundred numbers have reducible square roots. That’s only 37%, which is significantly lower than in the 40% or 39% of previous hundreds as this graphic illustrates:

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• 500 is a composite number.
• Prime factorization: 500 = 2 x 2 x 5 x 5 x 5, which can be written 500 = (2^2) x (5^3)
• The exponents in the prime factorization are 2 and 3. Adding one to each and multiplying we get (2 + 1)(3 + 1) = 3 x 4 = 12. Therefore 500 has exactly 12 factors.
• Factors of 500: 1, 2, 4, 5, 10, 20, 25, 50, 100, 125, 250, 500
• Factor pairs: 500 = 1 x 500, 2 x 250, 4 x 125, 5 x 100, 10 x 50, or 20 x 25
• Taking the factor pair with the largest square number factor, we get √500 = (√100)(√5) = 10√5 ≈ 22.36067977

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If you didn’t want to click, you can still see the horse race below, but the numbers from 401 to 500 will be much clearer if you click.

make animated gifs like this at MakeAGif

# 499 and Level 5

499 is the sum of a nice bunch of consecutive prime numbers:

• 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 = 499. That’s 17 consecutive primes.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-18

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• 499 is a prime number.
• Prime factorization: 499 is prime.
• The exponent of prime number 499 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 499 has exactly 2 factors.
• Factors of 499: 1, 499
• Factor pairs: 499 = 1 x 499
• 499 has no square factors that allow its square root to be simplified. √499 ≈ 22.3383079

How do we know that 499 is a prime number? If 499 were not a prime number, then it would be divisible by at least one prime number less than or equal to √499 ≈ 22.3383. Since 499 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 499 is a prime number.

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