A rhyme to help kids remember the even digits:

“0-2-4-6-8 being even is just great!”

That rhyme is often followed by another one to help kids remember the odd digits:

1-3-5-7-9 being odd is just fine!

Here’s a fun fact about the number 514:

514^3 = 135796744. OEIS.org tells us that 514 is the smallest number whose cube starts with all five of those odd digits in consecutive order.

514 is also the hypotenuse of the Pythagorean triple 64-510-514. What is the greatest common factor of those three numbers?

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-01

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• 514 is a composite number.
• Prime factorization: 514 = 2 x 257
• 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 514 has exactly 4 factors.
• Factors of 514: 1, 2, 257, 514
• Factor pairs: 514 = 1 x 514 or 2 x 257
• 514 has no square factors that allow its square root to be simplified. √514 ≈ 22.671568

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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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I found it difficult to find something interesting about the number 478 that wasn’t too difficult to understand, until I clicked on integernumber.com. It tells how to write 478 in several languages:

• English (EN): four hundred seventy-eight
• Spanish (ES): cuatrocientos setenta y ocho
• French (FR): quatre cent soixante-dix-huit
• German (DE): vierhundertachtundsiebzig
• Italian (IT): quattrocentosettantotto
• Hebrew (HE): ארבע-מאות שבעים ושמונה
• Indonesian (ID): empat ratus tujuh puluh delapan
• Russian (RU): четыреста семьдесят восемь
• Swedish (SV): fyrahundrasjutioåtta
• Turkish (TR): dört yüz yetmiş sekiz

In Hungarian 478 is négyszáz hetvennyolc.

If you know how to write 478 in any other language, please share it in the comments as some of my other readers have done.

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Today’s Level 5 puzzle may be a bit more difficult than usual, but give it a try anyway:

Print the puzzles or type the solution on this excel file: 10 Factors 2015-04-27

Here’s a little more about the number 478:

478 is palindrome 262 in BASE 14 because 2(196) + 6(14) + 2(1) = 478

478 can be written as the sum of four consecutive numbers:

118 + 119 + 120 + 121 = 478

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• 478 is a composite number.
• Prime factorization: 478 = 2 x 239
• 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 478 has exactly 4 factors.
• Factors of 478: 1, 2, 239, 478
• Factor pairs: 478 = 1 x 478 or 2 x 239
• 478 has no square factors that allow its square root to be simplified. √478 ≈ 21.86321

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471 is the sum of some consecutive prime numbers. One of my readers has listed those primes in the comments.

Print the puzzles or type the solution on this excel file:  12 Factors 2015-04-20

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• 471 is a composite number.
• Prime factorization: 471 = 3 x 157
• 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 471 has exactly 4 factors.
• Factors of 471: 1, 3, 157, 471
• Factor pairs: 471 = 1 x 471 or 3 x 471
• 471 has no square factors that allow its square root to be simplified. √471 ≈ 21.70253

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460 is the sum of consecutive prime numbers. Check the comments because one of my readers was able to find what those consecutive primes are.

Happy birthday to my son, Tim. I have two different cakes for you in this post. A cake puzzle and a simplified square root that uses the cake method that I’ve modified.

This puzzle will be included in an excel file of puzzles 12 Factors 2015-04-20.

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When we simplify square roots, we want to do as few divisions as possible. Since 60 can be evenly divided by perfect square 4, we know that 460 is also divisible by 4. Let’s use that fact to find its square root:

The quotient, 115, may be too large for us to know if it has any square factors. Since it isn’t divisible by 4, 9, or 25, let’s make a second layer to our cake as we divide it by its largest prime factor, 5.

Since the new quotient, 23, is a prime number, let’s revert back to the previous cake and take the square root of everything on the outside of the one layer cake: √460 = (√4)(√115) = 2√115.

• 460 is a composite number.
• Prime factorization: 460 = 2 x 2 x 5 x 23, which can be written 460 = (2^2) x 5 x 23
• 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 460 has exactly 12 factors.
• Factors of 460: 1, 2, 4, 5, 10, 20, 23, 46, 92, 115, 230, 460
• Factor pairs: 460 = 1 x 460, 2 x 230, 4 x 115, 5 x 92, 10 x 46, or 20 x 23
• Taking the factor pair with the largest square number factor, we get √460 = (√4)(√115) = 2√115 ≈ 21.4476

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Here’s the order the factors were found: