474 is the sum of some consecutive prime numbers which one of my readers listed in the comments. Try to figure out what they are yourself before checking there to see if you were right.

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

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• 474 is a composite number.
• Prime factorization: 474 = 2 x 3 x 79
• 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 474 has exactly 8 factors.
• Factors of 474: 1, 2, 3, 6, 79, 158, 237, 474
• Factor pairs: 474 = 1 x 474, 2 x 237, 3 x 158, or 6 x 79
• 474 has no square factors that allow its square root to be simplified. √474 ≈ 21.77154

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472 is divisible by 4 because 72 can be evenly divided by 4, and that means that √472 can be simplified. This is the method I use to reduce it:

First I divide 472 by 4 and get 118. If I recognize that 118 has no square factors, I simply can say √472 = (√4)(√118) = 2√118.

If I don’t recognize that 118 has no other square factors, I can divide it by 2 to get 59 which is a prime number and definitely has no square factors besides the number 1.

Then I take the square root of everything on the outside of the cake to get √472 = (√4)√(2 x 59) = (√4)(√118) = 2√118

Here is today’s puzzle:

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

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• 472 is a composite number.
• Prime factorization: 472 = 2 x 2 x 2 x 59, which can be written 472 = (2^3) x 59
• The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 472 has exactly 8 factors.
• Factors of 472: 1, 2, 4, 8, 59, 118, 236, 472
• Factor pairs: 472 = 1 x 472, 2 x 236, 4 x 118, or 8 x 59
• Taking the factor pair with the largest square number factor, we get √472 = (√4)(√118) = 2√118 ≈ 21.72556

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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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468 is the sum of some consecutive primes. One of my readers posted what those primes are in the comments.

468 is 3333 in base 5. (Thank you OEIS.org for that cool fact.) Here’s proof going from right to left using some easy division problems. (See 3333 at the bottom.)

And here’s proof going from left to right using some more difficult division problems. (See 3333 at the top.)

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The last two digits of 468 can be evenly divided by 4, so 468 is also divisible by 4.

468 is made from three consecutive even numbers so it is divisible by 3. Since the middle digit of the three consecutive even numbers is divisible by 3, we know that 468 can also be evenly divided by 9.

Let’s use those two facts to simplify the square root of 468 using the modified cake method. If you’re very confident in your ability to divide, you can make a one-layer cake and simply divide 468 by 36 to get 13. Then take the square root of everything on the outside of the cake and multiply them together: √468 = (√36)(√13) = 6√13

Many people will feel more comfortable making a two layer cake by dividing first by 4 and then by 9 as illustrated below:

Then to simplify √468, take the square root of everything on the outside of the cake and multiply those square roots together: √468 = (√4)(√9)(√13) = (2 x 3)(√13) = 6√13

You only need to know multiplication facts up to 12 x 12 to solve this factoring puzzle:

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

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• 468 is a composite number.
• Prime factorization: 468 = 2 x 2 x 3 x 3 x 13, which can be written 468 = (2^2) x (3^2) x 13
• The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 x 3 x 2 = 18. Therefore 468 has exactly 18 factors.
• Factors of 468: 1, 2, 3, 4, 6, 9, 12, 13, 18, 26, 36, 39, 52, 78, 117, 156, 234, 468
• Factor pairs: 468 = 1 x 468, 2 x 234, 3 x 156, 4 x 117, 6 x 78, 9 x 52, 12 x 39, 13 x 36 or 18 x 26
• Taking the factor pair with the largest square number factor, we get √468 = (√36)(√13) = 6√13 ≈ 21.6333

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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.

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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