# 477 and Level 4

4 + 7 + 7 = 18, a multiple of 9, so 477 can be evenly divided by 9, and 477 is the third number in a row whose square root can be simplified.

To find the square root we only need to make a one-layer cake before we take the square root of everything on the outside of the cake: √477 = (√9)(√53) = 3√53

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

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• 477 is a composite number.
• Prime factorization: 477 = 3 x 3 x 53, which can be written 477 = (3^2) x 53
• The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 477 has exactly 6 factors.
• Factors of 477: 1, 3, 9, 53, 159, 477
• Factor pairs: 477 = 1 x 477, 3 x 159, or 9 x 53
• Taking the factor pair with the largest square number factor, we get √477 = (√9)(√53) = 3√53 ≈ 21.84033

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# 476 and Level 3

76 can be evenly divided by 4 so 476 is also divisible by 4. Since 476 has a square factor, √476 can be simplified. Reducing square roots is easy if you make a cake: First divide 476 by its smallest square factor, 4, to get 119, a number that doesn’t have any square factors. Dividing 119 by its smallest prime factor is optional, but can help you see that 119, indeed, has no square factors. Last of all, take the square root of everything on the outside of the cake and multiply them together. √476 = (√4)(√(7 x 17) = (√4)(√119) = 2√119.

Here is today’s factoring puzzle:

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

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• 476 is a composite number.
• Prime factorization: 476 = 2 x 2 x 7 x 17, which can be written 476 = (2^2) x 7 x 17
• 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 476 has exactly 12 factors.
• Factors of 476: 1, 2, 4, 7, 14, 17, 28, 34, 68, 119, 238, 476
• Factor pairs: 476 = 1 x 476, 2 x 238, 4 x 119, 7 x 68, 14 x 34, or 17 x 28
• Taking the factor pair with the largest square number factor, we get √476 = (√4)(√119) = 2√119 ≈ 21.81742

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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.

# 475 and Level 2

Numbers ending in 25, 50, 75, and 00 can be evenly divided by 25 so their square roots can be simplified. What is 475 divided by 25? You can find the answer by asking yourself, “If I had \$4.75 in quarters, how many quarters would I have?”

If you had \$5.00 in quarters, you would have 20 quarters, so \$4.75 is 19 quarters. Thus √475 = (√25)(√19) = 5√19.

What is the greatest common factor of the numbers in this Pythagorean triple, 133-456-475? Let’s look at the difference between 456 and the other two numbers:

• 456 – 133 = 323
• 475 – 456 = 19
• Since 19 is the smaller difference, the greatest common factor has to be a factor of 19, and 19 just happens to be a factor of all three numbers.
• 7-24-25 multiplied by 19 is 133-456-475.

What is the greatest common factor of the numbers in Pythagorean triple, 285-380-475? Find the difference between 380 and the other two numbers:

• 380 – 285 = 95
• 475 – 380 = 95
• Since the differences are equal, we know that this Pythagorean triple is just 3-4-5 multiplied by 95, and the greatest common factor is 95.

Here is today’s factoring puzzle:

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

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• 475 is a composite number.
• Prime factorization: 475 = 5 x 5 x 19, which can be written 475 = (5^2) x 19
• The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 475 has exactly 6 factors.
• Factors of 475: 1, 5, 19, 25, 95, 475
• Factor pairs: 475 = 1 x 475, 5 x 95, or 19 x 25
• Taking the factor pair with the largest square number factor, we get √475 = (√25)(√19) = 5√19 ≈ 21.79449

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# 474 and Level 1

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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# Just a Little 473 Cake

Today is our wedding anniversary. Today I tried to spend my time with my husband instead of the computer. Hence this is a very short and plain post.

The middle digit of 473 equals the sum of the other two digits which means that 473 can be evenly divided by 11.

Five years from now I could make one of these cakes for our anniversary!

473 is the sum of some consecutive prime numbers two different ways. See if you can find them yourself. Then check the comments to see if you were right.

• 473 is a composite number.
• Prime factorization: 473 = 11 x 43
• 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 473 has exactly 4 factors.
• Factors of 473: 1, 11, 43, 473
• Factor pairs: 473 = 1 x 473 or 11 x 43
• 473 has no square factors that allow its square root to be simplified. √473 ≈ 21.74856

# How to Simplify √472

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 and Level 5

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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# 470 Greatest Common Factors of Pythagorean Triples.

• 470 is a composite number.
• Prime factorization: 470 = 2 x 5 x 47
• 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 470 has exactly 8 factors.
• Factors of 470: 1, 2, 5, 10, 47, 94, 235, 470
• Factor pairs: 470 = 1 x 470, 2 x 235, 5 x 94, or 10 x 47
• 470 has no square factors that allow its square root to be simplified. √470 ≈ 21.67948

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470 is the hypotenuse of the non-primitive Pythagorean triple 282-376-470. What is the greatest common factor of those three numbers?

The greatest common factor will always be a factor of the smallest of the three numbers, but it will also be a factor of the smallest difference between the three numbers as well. Let’s find those differences. Note: the difference between the 282 and 470 will not be the smallest difference so there is no need to find that one. We only need to find these two differences:

In the case of this Pythagorean triple the differences are equal to each other which means that the difference, 94*, is also the greatest common factor of the three numbers! Go ahead and try dividing each number in the triple by 94. You will discover that this Pythagorean triple is just 3-4-5 multiplied by 94.

*This statement is only true of Pythagorean triples. For example the following numbers also have differences of 94, but the greatest common factor is not 94, but a factor of 94:

1. The greatest common factor of 283-377-471 is 1.
2. The greatest common factor of 284-378-472 is 2
3. The greatest common factor of 329-423-517 is 47

Mathchat has written an excellent post on finding the greatest common factor of three or more numbers that can be used for all integers in general.

But as far as Pythagorean triples are concerned, anytime the corresponding differences of a Pythagorean triple are equal to each other, then that Pythagorean triple is just 3-4-5 multiplied by the difference. There are an infinite number of such triples, and 282-376-470 is just one of them.

Now remember there is an infinite number of primitive Pythagorean triples, and every one of those triples can be multiplied by each of the infinitely many counting numbers. A graphic like the one above could be made for every primitive triple followed by each of its multiples. For example 5-12-13, 10-24-26, 15- 36-39, etc. would be another infinite series of Pythagorean triples.

You could say the total number of Pythagorean triples equals infinity times infinity!

# 469 and Level 4

469 is the short leg in the Pythagorean triple 469-1608-1675. What is the greatest common factor of those three numbers? Hint: Don’t let the larger numbers scare you; the greatest common factor is a factor of 469, the smallest of those three numbers, and its factors are listed below the puzzle.

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

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• 469 is a composite number.
• Prime factorization: 469 = 7 x 67
• 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 469 has exactly 4 factors.
• Factors of 469: 1, 7, 67, 469
• Factor pairs: 469 = 1 x 469 or 7 x 67
• 469 has no square factors that allow its square root to be simplified. √469 ≈ 21.6564

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# 468 and Level 3

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 stetson.edu 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.