533 is the sum of consecutive primes two different ways: 173 + 179 + 181 = 533 = 101 + 103 + 107 + 109 + 113.

533 = (23^2) + (2^2), and 533 = (22^2) + (7^2)

533 is the hypotenuse of four Pythagorean triples. Some of the triples have a greatest common factor greater than one, and the rest are primitive. Which are which?

• 92-525-533
• 117-520-533
• 205-492-533
• 308-435-533

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-22

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• 533 is a composite number.
• Prime factorization: 533 = 13 x 41
• 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 533 has exactly 4 factors.
• Factors of 533: 1, 13, 41, 533
• Factor pairs: 533 = 1 x 533 or 13 x 41
• 533 has no square factors that allow its square root to be simplified. √533 ≈ 23.08679

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5 + 3 + 1 = 9 so 531 is divisible by 3 and by 9. Because it can be evenly divided by 9, it’s square root can be reduced.

531 ÷ 9 = 59, a prime number which obviously has no square factors. Thus √531 = (√9)(√59) = 3√59 in its most reduced form.

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-22

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• 531 is a composite number.
• Prime factorization: 531 = 3 x 3 x 59, which can be written 531 = (3^2) x 59
• 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 531 has exactly 6 factors.
• Factors of 531: 1, 3, 9, 59, 177, 531
• Factor pairs: 531 = 1 x 531, 3 x 177, or 9 x 59
• Taking the factor pair with the largest square number factor, we get √531 = (√9)(√59) = 3√59 ≈ 23.04343724

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The first three perfect numbers are 6, 28, and 496. Their sum is 530.

Can you find the greatest common factors for each of these Pythagorean triples?

• 46-528-530
• 192-494-530
• 280-450-530
• 318-424-530

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-22

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• 530 is a composite number.
• Prime factorization: 530 = 2 x 5 x 53
• 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 530 has exactly 8 factors.
• Factors of 530: 1, 2, 5, 10, 53, 106, 265, 530
• Factor pairs: 530 = 1 x 530, 2 x 265, 5 x 106, or 10 x 53
• 530 has no square factors that allow its square root to be simplified. √530 ≈ 23.02172886644

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528 feet is one tenth of a mile. Joseph Nebus has written a fascinating post on WHY a mile is 5280 feet. Factoring that number is part of the reason.

Since 528 = (32 x 33)/ 2, we know that 528 is the 32nd triangular number.

Thus, 1 + 2 + 3 + 4 + . . . . . + 29 + 30 + 31 + 32 = 528.

528 can be written as the difference of two squares six different ways:

How did I find them? Choose a factor pair where both factors are even or both factors are odd. Average the two factors and that will be the first number to be squared. Square it and subtract 528. Take the square root of the result. That number is the second number in the difference of two squares.

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

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• 528 is a composite number.
• Prime factorization: 528 = 2 x 2 x 2 x 2 x 3 x 11, which can be written 528 = (2^4) x 3 x 11
• The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 x 2 x 2 = 20. Therefore 528 has exactly 20 factors.
• Factors of 528: 1, 2, 3, 4, 6, 8, 11, 12, 16, 22, 24, 33, 44, 48, 66, 88, 132, 176, 264, 528
• Factor pairs: 528 = 1 x 528, 2 x 264, 3 x 176, 4 x 132, 6 x 88, 8 x 66, 11 x 48, 12 x 44, 16 x 33 or 22 x 24
• Taking the factor pair with the largest square number factor, we get √528 = (√16)(√33) = 4√33 ≈ 22.97825

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527 is the sum of consecutive primes including all the prime numbers in its prime factorization. Can you list those consecutive primes?

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

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• 527 is a composite number.
• Prime factorization: 527 = 17 x 31
• 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 527 has exactly 4 factors.
• Factors of 527: 1, 17, 31, 527
• Factor pairs: 527 = 1 x 527 or 17 x 31
• 527 has no square factors that allow its square root to be simplified. √527 ≈ 22.95648

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