Today’s Puzzle:

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

Something Cool about √504:

I’ve made a little cake to simplify √504:

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, a pretty cool-looking approximation!

Factors of 504:

• 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

Sum-Difference Puzzle:

504 has twelve factor pairs. One of the factor pairs adds up to 65, and a different one subtracts to 65. If you can identify those factor pairs, then you can solve this puzzle!

A Puzzle about the Number 504:

The numbers 360, 420, 480, and 504 have something 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 the same special something that those other three numbers have. Can you figure out what it is? Hint: It has something to do with the number 24.

Factors for Today’s Puzzle:

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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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 OEIS.org.

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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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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As I searched for something interesting yet understandable to say about the number 498, I learned that 498 is the third 167-gonal number when I looked at virtuescience.com/polygonal-numbers. I also found out that “Charlie Eppes” has a blog called Numbers that features graphic representations of the number 498 and a lot of other numbers, too.

You may find puzzle #498 to be a little trickier than most Level 4 puzzles, but I know you can still meet the challenge.

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

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• 498 is a composite number.
• Prime factorization: 498 = 2 x 3 x 83
• 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 498 has exactly 8 factors.
• Factors of 498: 1, 2, 3, 6, 83, 166, 249, 498
• Factor pairs: 498 = 1 x 498, 2 x 249, 3 x 166, or 6 x 83
• 498 has no square factors that allow its square root to be simplified. √498 ≈ 22.3159136

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• 6 is the first perfect number because 1 + 2 + 3 = 6. Also notice that (2^1)(2^2 – 1) = 2 x 3 = 6.
• 28 is the second perfect number because 1 + 2 + 4 + 7 + 14 = 28. Hmm… (2^2)(2^3 – 1) = 4 x 7 = 28.
• (2^3)(2^4 – 1) = 8 x 15 = 120 is NOT a perfect number because 1 + 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 + 15 + 20 + 24 + 30 + 40 + 60 = 240 which is two times what it needs to be.
• 496 is (2^4)(2^5 – 1) = 16 x 31. So why is 496 the third perfect number? Everything you need to know to figure out the answer to that question can be found somewhere in this post.
• 2016 is (2^5)(2^6 – 1) = 32 x 63, and 2016 is also NOT a perfect number.

6, 28, and 496 are all triangular numbers as well as hexagonal numbers, but 120 and 2016 can also make that claim.

The clues in yesterday’s Find the Factors puzzle were all perfect squares. Today’s puzzle is only a little more difficult. You can solve it, too!

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

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• 496 is a composite number.
• Prime factorization: 496 = 2 x 2 x 2 x 2 x 31, which can be written 496 = (2^4) x 31
• The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 x 2 = 10. Therefore 496 has exactly 10 factors.
• Factors of 496: 1, 2, 4, 8, 16, 31, 62, 124, 248, 496
• Factor pairs: 496 = 1 x 496, 2 x 248, 4 x 124, 8 x 62, or 16 x 31
• Taking the factor pair with the largest square number factor, we get √496 = (√16)(√31) = 4√31 ≈ 22.271057

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495 has an almost magical property and is known as the Kaprekar constant for 3 digit numbers. What does that mean?

Take any 3 digit number that has at least 2 different digits. Write the digits from greatest to least to create a new 3-digit number. From that number subtract the same digits written in reverse order. Repeat this process, and you will get the number 495 in no more than seven iterations. This graphic shows this process applied to the number 101.

The following puzzle can easily transform into a multiplication table if you first find all the square factors of the given clues. Go ahead give it a try!

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

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• 495 is a composite number.
• Prime factorization: 495 = 3 x 3 x 5 x 11, which can be written 495 = (3^2) x 5 x 11
• 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 495 has exactly 12 factors.
• Factors of 495: 1, 3, 5, 9, 11, 15, 33, 45, 55, 99, 165, 495
• Factor pairs: 495 = 1 x 495, 3 x 165, 5 x 99, 9 x 55, 11 x 45, or 15 x 33
• Taking the factor pair with the largest square number factor, we get √495 = (√9)(√55) = 3√55 ≈ 22.248595

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