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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493 is the sum of some consecutive prime numbers. Check the comments to see who posts those consecutive primes first, one of my readers or me.

David Radcliffe posted this 493-related gif on twitter:

All 493 ways to tile an 8×8 square (corners removed) using all 12 pentominoes. pic.twitter.com/hTOlmAe5va

— David Radcliffe (@daveinstpaul) January 3, 2017

//platform.twitter.com/widgets.js

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-11

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• 493 is a composite number.
• Prime factorization: 493 = 17 x 29
• 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 493 has exactly 4 factors.
• Factors of 493: 1, 17, 29, 493
• Factor pairs: 493 = 1 x 493 or 17 x 29
• 493 has no square factors that allow its square root to be simplified. √493 ≈ 22.203603

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492 is the sum of some consecutive prime numbers (actually two different ways: see the comments). I’ll list those primes in the comments in about a week unless somebody else beats me to it. (abyssbrain beat me to it.)

Simplifying the square root of 492 is as easy as 1-2-3:

92 can be evenly divided by 4, so 492 is also divisible by 4. If I wanted to find √492, I would make a little cake and divide 492 by 4.

123 is not divisible by 4 or by 9, but it is divisible by 3 so I will do that division next and get a quotient of 41, a prime number. Now I will take the square root of everything on the outside of the cake and multiply them together: √492 = (√4)(√3)√(41) = (√4)(√123) =2√123

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-11

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• 492 is a composite number.
• Prime factorization: 492 = 2 x 2 x 3 x 41, which can be written 492 = (2^2) x 3 x 41
• 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 492 has exactly 12 factors.
• Factors of 492: 1, 2, 3, 4, 6, 12, 41, 82, 123, 164, 246, 492
• Factor pairs: 492 = 1 x 492, 2 x 246, 3 x 164, 4 x 123, 6 x 82, or 12 x 41
• Taking the factor pair with the largest square number factor, we get √492 = (√4)(√123) = 2√123 ≈ 22.181073

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491 is the sum of consecutive primes two different ways. Check the comments to see if any of my readers list what those consecutive primes are.

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-11

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• 491 is a prime number.
• Prime factorization: 491 is prime.
• The exponent of prime number 491 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 491 has exactly 2 factors.
• Factors of 491: 1, 491
• Factor pairs: 491 = 1 x 491
• 491 has no square factors that allow its square root to be simplified. √491 ≈ 22.1585198

How do we know that 491 is a prime number? If 491 were not a prime number, then it would be divisible by at least one prime number less than or equal to √491 ≈ 22.159. Since 491 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 491 is a prime number.

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