A Multiplication Based Logic Puzzle

Archive for April, 2015

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.

477 cake

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

477 Puzzle

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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477 Logic

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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.

476 cake

Here is today’s factoring puzzle:

476 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.

476 Factors

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.

475 cake

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:

475 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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475 Factors

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.

474 Puzzle

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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474 Factors

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.

472 cake

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:

472 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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472 Logic

471 and Level 5

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

471 Puzzle

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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471 Logic

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