A Multiplication Based Logic Puzzle

Archive for May, 2015

508 A Mathematical and Biblical Truth: 2 > 1

Two Is Better Than One Because...

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
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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 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:

507 2-layer 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:

507 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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507 Logic

506 and Level 5

506 is the eleventh 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.

And 506 = (11 x 12 x 23)/6

506 Puzzle

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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506 Logic

505 and Level 4

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

505 = 10C5 + 10C0 + 10C5.

505 Puzzle

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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505 Logic

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?

504 Puzzle

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:

504 cake

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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504 Factors

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.

503 Puzzle

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 and cannot be factored.
  • 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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503 Factors

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.

502 Puzzle

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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502 Factors

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