4 + 8 = 12, a multiple of 3, so 483 can be evenly divided by 3. (It wasn’t necessary to add 4 + 8 + 3 because we already know 3 is divisible by 3.)

We can easily see that 483 is divisible by 7 if we apply the divisibility trick for 7. Separate the last number, 3, from the rest, double it, and subtract the double from the remaining numbers: 48 – (3 x 2) = 48 – 6 = 42, which is a multiple of 7. Thus 483 is divisible by 7.

Since 483 is 21 x 23, we can predict that 484 will be 22 x 22.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-04

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• 483 is a composite number.
• Prime factorization: 483 = 3 x 7 x 23
• 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 483 has exactly 8 factors.
• Factors of 483: 1, 3, 7, 21, 23, 69, 161, 483
• Factor pairs: 483 = 1 x 483, 3 x 161, 7 x 69, or 21 x 23
• 483 has no square factors that allow its square root to be simplified. √483 ≈ 21.97726

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Can you find the greatest common factor of the three numbers in this Pythagorean triple 156-455-481? Or in Pythagorean triple 185-444-481?

481 is also the hypotenuse in two primitive Pythagorean triples: 319-360-481 and 31-480-481. Since they are primitive triples, their greatest common factor is 1.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-04

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• 481 is a composite number.
• Prime factorization: 481 = 13 x 37
• 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 481 has exactly 4 factors.
• Factors of 481: 1, 13, 37, 481
• Factor pairs: 481 = 1 x 481 or 13 x 37
• 481 has no square factors that allow its square root to be simplified. √481 ≈ 21.93171

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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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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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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.

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 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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