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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490 is seventy times seven. A famous Bible verse mentioning this number is Matthew 18: 21 – 22.

This is the King James Version: “Then came Peter to him, and said, Lord, how oft shall my brother sin against me, and I forgive him? till seven times? Jesus saith unto him, I say not unto thee, Until seven times: but, Until seventy times seven.”

Most other Bible translation also have seventy times seven, but a few translations have only seventy seven. You can compare several different translations of Matthew 18: 22 here. Regardless of the translation, forgiveness is much more important than keeping count especially since all of us need to be forgiven as well.

To find √490 first notice that 490 can be evenly divided by 2, but not by 4. It also cannot be evenly divided by 9. It is divisible by 5, but since it doesn’t end in 25, 50, 75, or 00, it can only be divided by 5 once. Since it is divisible by 10, and dividing by 10 is so easy, I would make this one-layer cake:

Then I would take the square root of everything on the outside of the cake and multiply together to get √490 = (√10)(√49) = 7√10.

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

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• 490 is a composite number.
• Prime factorization: 490 = 2 x 5 x 7 x 7, which can be written 490 = 2 x 5 x (7^2)
• The exponents in the prime factorization are 1, 1, and 2. Adding one to each and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 x 2 x 3 = 12. Therefore 490 has exactly 12 factors.
• Factors of 490: 1, 2, 5, 7, 10, 14, 35, 49, 70, 98, 245, 490
• Factor pairs: 490 = 1 x 490, 2 x 245, 5 x 98, 7 x 70, 10 x 49, or 14 x 35
• Taking the factor pair with the largest square number factor, we get √490 = (√49)(√10) = 7√10 ≈ 22.135943

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

 

 

 

489 is the ninth octahedral number.  Its square root starts out in quite a memorable way: √489 ≈ 22.113344, but after those six decimal places, it looks as irrational as can be.

 

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

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• 489 is a composite number.
• Prime factorization: 489 = 3 x 163
• 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 489 has exactly 4 factors.
• Factors of 489: 1, 3, 163, 489
• Factor pairs: 489 = 1 x 489 or 3 x 163
• 489 has no square factors that allow its square root to be simplified. √489 ≈ 22.113344

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