466 is an interesting number whose factors are listed below today’s puzzle.

Thanks to OEIS.org, I know that 466 is 1234 in base 7. There are two different ways to change 466 from base 10 to base 7.

The first way gives the answer at the top of all the division problems and has you working from left to right. The division problems may be more difficult because division by 7 cubed and 7 squared are required, but the concept of what is happening is fairly easy to understand.

For the second way, each of the division problems is quite easy to do. However working from right to left and finding the answer at the bottom of all the problems may be confusing to some people.

Changing a number from base ten to base seven can be a bit of a challenge. However, today’s Find the Factors puzzle is as easy as they get. Every person who has learned how to multiply can find the factors for this puzzle and then make the puzzle work like a multiplication table:

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

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Here is the factoring information for 466:

• 466 is a composite number.
• Prime factorization: 466 = 2 x 233
• 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 466 has exactly 4 factors.
• Factors of 466: 1, 2, 233, 466
• Factor pairs: 466 = 1 x 466 or 2 x 233
• 466 has no square factors that allow its square root to be simplified. √466 ≈ 21.58703

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464 is the hypotenuse of Pythagorean triple 320-336-464. Can you figure out what is the greatest common factor of those three numbers? Hint: it has to be an even factor of 320 because that is the smallest of those three even numbers.

The logic needed to begin this Level 6 puzzle shouldn’t be too difficult to discover.

Print the puzzles or type the solution on this excel file:  10 Factors 2015-04-13

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If I wanted to simplify √464, I would first notice that its last two digits, 64, are divisible by 4, so 464 also is divisible by 4. I would make a little cake like this:

464 ÷ 4 = 116. Guess what? 116 is also divisible by 4 because 16 is divisible by 4. I would make another layer for my cake like this:

29 is a prime number so my cake is finished. Now to simplify √464, I would just take the square root of everything on the outside of the cake and multiply them together.

√464 = (√4)(√4)(√29) = 4√29

• 464 is a composite number.
• Prime factorization: 464 = 2 x 2 x 2 x 2 x 29, which can be written 464 = (2^4) x 29
• 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 464 has exactly 10 factors.
• Factors of 464: 1, 2, 4, 8, 16, 29, 58, 116, 232, 464
• Factor pairs: 464 = 1 x 464, 2 x 232, 4 x 116, 8 x 58, or 16 x 29
• Taking the factor pair with the largest square number factor, we get √464 = (√16)(√29) = 4√29 ≈ 21.5407

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462 is the sum of consecutive prime numbers two different ways. Check the comments to see what those ways are.

Divisibility tricks:

• 462 is even, so it is divisible by 2.
• The sum of the odd numbered digits, 4 + 2 is 6, which is the 2nd digit, so 462 is divisible by 11.
• Since both of those 6’s above are divisible by 3, then 462 is divisible by 3.
• Separate the last digit from the rest and double it. 462 → 46 and 2; Doubling 2, gives us 4. Now subtract that 4 from the remaining digits: 46 – 4 = 42 which is divisible by 7, so 462 is divisible by 7.

Since 462 = 21 × 22, we know that it is two times the 21st triangular number, and it is the sum of the first 21 even numbers.

• 2(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21) = 462
• 2 + 4 +  6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30 + 32 + 34 + 36 + 38 + 40 + 42 = 462

Print the puzzles or type the solution on this excel file:  10 Factors 2015-04-13

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• 462 is a composite number.
• Prime factorization: 462 = 2 x 3 x 7 x 11
• The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 x 2 = 16. Therefore 462 has exactly 16 factors.
• Factors of 462: 1, 2, 3, 6, 7, 11, 14, 21, 22, 33, 42, 66, 77, 154, 231, 462
• Factor pairs: 462 = 1 x 462, 2 x 231, 3 x 154, 6 x 77, 7 x 66, 11 x 42, 14 x 33, or 21 x 22
• 462 has no square factors that allow its square root to be simplified. √462 ≈ 21.4942

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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:

458 = (13^2) + (17^2). It is the hypotenuse of this Pythagorean triple: 120-442-458.

Print the puzzles or type the solution on this excel file:  10 Factors 2015-04-13

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• 458 is a composite number.
• Prime factorization: 458 = 2 x 229
• 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 458 has exactly 4 factors.
• Factors of 458: 1, 2, 229, 458
• Factor pairs: 458 = 1 x 458 or 2 x 229
• 458 has no square factors that allow its square root to be simplified. √458 ≈ 21.4009

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