552 is the sum of all the prime numbers from 37 to 73.

It is also the sum of all the prime numbers from 79 to 103.

That means if you add up all the prime numbers from 37 to 103, you’ll get 1104 which is 2 x 552.

Can you list all those prime numbers?

Print the puzzles or type the solution on this excel file: 12 Factors 2015-07-13

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I’ve modified the cake method to help me find √552.  My philosophy is that there is no reason to completely break up 552 to find its square root when we’re just going to have to put it mostly back together again to give the final answer. About 83% of numbers whose square roots can be reduced are divisible by 4 and/or 9. It makes sense to check to see if 552 is divisible by either of those numbers first.

552 can be evenly divided by 4 because its last two digits are divisible by 4. That means √552 can be reduced. Let’s find 552 ÷ 4.

The quotient is 138 which is even but not divisible by 4. Let’s check to see if it is divisible by nine: 1 + 3 + 8 = 12. It is divisible by 3, but not by 9.

Since 138 is even and divisible by 3, it is divisible by 6. It is easier to divide by 6 once than it is to divide first by 2 and then by 3. Dividing by 6 makes our division cake look like this:

Since 23 is a prime number, we are done with the division process. Taking the square root of everything on the outside of the cake we get √552 = (√4)(√6)(√23) = (√4)(√138) = 2√138.

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• 552 is a composite number.
• Prime factorization: 552 = 2 x 2 x 2 x 3 x 23, which can be written 552 = (2^3) x 3 x 23
• The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 552 has exactly 16 factors.
• Factors of 552: 1, 2, 3, 4, 6, 8, 12, 23, 24, 46, 69, 92, 138, 184, 276, 552
• Factor pairs: 552 = 1 x 552, 2 x 276, 3 x 184, 4 x 138, 6 x 92, 8 x 69, 12 x 46, or 23 x 24
• Taking the factor pair with the largest square number factor, we get √552 = (√4)(√138) = 2√138 ≈ 23.49468

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24 is divisible by 4 and that means 524 is also divisible by 4.

Numbers that are divisible by 4 can have their square roots reduced. 524 ÷ 4 = 131, a prime number whose only square factor is 1, so √524 = (√4)(√131) = 2√131.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-15

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• 524 is a composite number.
• Prime factorization: 524 = 2 x 2 x 131, which can be written 524 = (2^2) x 131
• 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 524 has exactly 6 factors.
• Factors of 524: 1, 2, 4, 131, 262, 524
• Factor pairs: 524 = 1 x 524, 2 x 262, or 4 x 131
• Taking the factor pair with the largest square number factor, we get √524 = (√4)(√131) = 2√131 ≈ 22.891046

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5 – 1 + 7 = 11 so 517 can be evenly divided by 11.

517 = 97 + 101 + 103 + 107 + 109 which is all the prime numbers between 90 and 112.

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-08

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• 517 is a composite number.
• Prime factorization: 517 = 11 x 47
• 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 517 has exactly 4 factors.
• Factors of 517: 1, 11, 47, 517
• Factor pairs: 517 = 1 x 517 or 11 x 47
• 517 has no square factors that allow its square root to be simplified. √517 ≈ 22.737634

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510 is the sum of consecutive prime numbers three different ways.

1. 510 = 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79
2. 510 = 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71
3. 510 = 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67

510 is the hypotenuse of four Pythagorean triples, none of which are primitive:

1. 78-504-510
2. 216-462-510
3. 240-450-510
4. 306-408-510

Two of those triples have the same greatest common factor. Can you identify which ones they are?

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-01

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• 510 is a composite number.
• Prime factorization: 510 = 2 x 3 x 5 x 17
• 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 510 has exactly 16 factors.
• Factors of 510: 1, 2, 3, 5, 6, 10, 15, 17, 30, 34, 51, 85, 102, 170, 255, 510
• Factor pairs: 510 = 1 x 510, 2 x 255, 3 x 170, 5 x 102, 6 x 85, 10 x 51, 15 x 34, or 17 x 30
• 510 has no square factors that allow its square root to be simplified. √510 ≈ 22.58317958

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503 = (2^3) + (3^3) + (5^3) + (7^3) which is the sum of the cubes of the first four prime numbers. 503 is the smallest prime number that is the sum of consecutive cubes of prime numbers.

503 is also the sum of three consecutive prime numbers: 163, 167, and 173.

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-25

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• 503 is a prime number.
• Prime factorization: 503 is prime.
• The exponent of prime number 503 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 503 has exactly 2 factors.
• Factors of 503: 1, 503
• Factor pairs: 503 = 1 x 503
• 503 has no square factors that allow its square root to be simplified. √503 ≈ 22.42766

How do we know that 503 is a prime number? If 503 were not a prime number, then it would be divisible by at least one prime number less than or equal to √503 ≈ 22.4. Since 503 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 503 is a prime number.

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• 6 is the first perfect number because 1 + 2 + 3 = 6. Also notice that (2^1)(2^2 – 1) = 2 x 3 = 6.
• 28 is the second perfect number because 1 + 2 + 4 + 7 + 14 = 28. Hmm… (2^2)(2^3 – 1) = 4 x 7 = 28.
• (2^3)(2^4 – 1) = 8 x 15 = 120 is NOT a perfect number because 1 + 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 + 15 + 20 + 24 + 30 + 40 + 60 = 240 which is two times what it needs to be.
• 496 is (2^4)(2^5 – 1) = 16 x 31. So why is 496 the third perfect number? Everything you need to know to figure out the answer to that question can be found somewhere in this post.
• 2016 is (2^5)(2^6 – 1) = 32 x 63, and 2016 is also NOT a perfect number.

6, 28, and 496 are all triangular numbers as well as hexagonal numbers, but 120 and 2016 can also make that claim.

The clues in yesterday’s Find the Factors puzzle were all perfect squares. Today’s puzzle is only a little more difficult. You can solve it, too!

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-18

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• 496 is a composite number.
• Prime factorization: 496 = 2 x 2 x 2 x 2 x 31, which can be written 496 = (2^4) x 31
• 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 496 has exactly 10 factors.
• Factors of 496: 1, 2, 4, 8, 16, 31, 62, 124, 248, 496
• Factor pairs: 496 = 1 x 496, 2 x 248, 4 x 124, 8 x 62, or 16 x 31
• Taking the factor pair with the largest square number factor, we get √496 = (√16)(√31) = 4√31 ≈ 22.271057

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489 is the ninth octahedral number.  Its square root starts out in quite a memorable way: √489 ≈ 22.113344, but after those six decimal places, it looks as irrational as can be.

 

Print the puzzles or type the solution on this excel file: 10 Factors 2015-05-11

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• 489 is a composite number.
• Prime factorization: 489 = 3 x 163
• 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 489 has exactly 4 factors.
• Factors of 489: 1, 3, 163, 489
• Factor pairs: 489 = 1 x 489 or 3 x 163
• 489 has no square factors that allow its square root to be simplified. √489 ≈ 22.113344

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