Here are two divisibility rules applied to 627 that give a positive result:

• 6 + 2 + 7 = 15, a multiple of 3, so 627 is divisible by 3.
• 6 – 2 + 7 = 11, so 627 is divisible by 11.

627 can be expressed as the sum of consecutive counting numbers 4 different ways. The numbers in bold are in the middle of each sum:

• 208 + 209 + 210 = 627
• 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 = 627
• 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 = 627
• 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11+ 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 = 627

The logic on this Level 5 puzzle gets a bit complicated right when it’s almost done:

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-21

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• 627 is a composite number.
• Prime factorization: 627 = 3 x 11 x 19
• 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 627 has exactly 8 factors.
• Factors of 627: 1, 3, 11, 19, 33, 57, 209, 627
• Factor pairs: 627 = 1 x 627, 3 x 209, 11 x 57, or 19 x 33
• 627 has no square factors that allow its square root to be simplified. √627 ≈ 25.039968.

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Can 622 be written as the sum of consecutive numbers?

622 is greater than 6 and is divisible by 2, but not by 4, thus it can be expressed as the sum of 4 consecutive numbers:

154 + 155 + 156 + 157 = 622

622  is the sum of the sixteen prime numbers from 11 to 71.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-21

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• 622 is a composite number.
• Prime factorization: 622 = 2 x 311
• 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 622 has exactly 4 factors.
• Factors of 622: 1, 2, 311, 622
• Factor pairs: 622 = 1 x 622 or 2 x 311
• 622 has no square factors that allow its square root to be simplified. √622 ≈ 24.9399.

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620 is the sum of the four prime numbers from 149 to 163, and it is also the sum of the eight prime numbers from 61 to 97.

620 is the hypotenuse of the Pythagorean triple 372-496-620. What is the greatest common factor of those three numbers?

Today’s puzzle may be a little more difficult than most Level 5 puzzles, but go ahead, embrace the challenge!

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-14

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• 620 is a composite number.
• Prime factorization: 620 = 2 x 2 x 5 x 31, which can be written 620 = (2^2) x 5 x 31
• 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 620 has exactly 12 factors.
• Factors of 620: 1, 2, 4, 5, 10, 20, 31, 62, 124, 155, 310, 620
• Factor pairs: 620 = 1 x 620, 2 x 310, 4 x 155, 5 x 124, 10 x 62, or 20 x 31
• Taking the factor pair with the largest square number factor, we get √620 = (√4)(√155) = 2√155 ≈ 24.899799

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619  is the 12^th strobogrammatic number because it looks like the same number when it is turned upside down. The first 4 strobogrammatic prime numbers are 11, 101, 181, and 619.

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-14

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• 619 is a prime number.
• Prime factorization: 619 is prime.
• The exponent of prime number 619 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 619 has exactly 2 factors.
• Factors of 619: 1, 619
• Factor pairs: 619 = 1 x 619
• 619 has no square factors that allow its square root to be simplified. √619 ≈ 24.87971.

How do we know that 619 is a prime number? If 619 were not a prime number, then it would be divisible by at least one prime number less than or equal to √619 ≈ 24.9. Since 619 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 619 is a prime number.

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618 is even so it is obviously divisible by 2. Is it divisible by 3?

Yes, 618 has the same digits as 16 & 8. Since 16 is 8 doubled, 618 is divisible by 3.

618  is the sum of consecutive prime numbers 307 and 311.

618 can be written as the sum of 4 consecutive numbers because 618 is greater than 6 and is divisible by 2, but not by 4. Thus 153 + 154 + 155 + 156 = 618.

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-14

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• 618 is a composite number.
• Prime factorization: 618 = 2 x 3 x 103
• 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 618 has exactly 8 factors.
• Factors of 618: 1, 2, 3, 6, 103, 206, 309, 618
• Factor pairs: 618 = 1 x 618, 2 x 309, 3 x 206, or 6 x 103
• 618 has no square factors that allow its square root to be simplified. √618 ≈ 24.8596.

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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. (None of the factors are greater than 10.)  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.

I learned from OEIS.org that 614 is the smallest integer that can be expressed as the sum of 3 squares 9 different ways. I decided to see if I could find those 9 ways. Here they are:

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-07

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• 614 is a composite number.
• Prime factorization: 614 = 2 x 307
• 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 614 has exactly 4 factors.
• Factors of 614: 1, 2, 307, 614
• Factor pairs: 614 = 1 x 614 or 2 x 307
• 614 has no square factors that allow its square root to be simplified. √614 ≈ 24.77902

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608 is the sum of 19 consecutive numbers:

23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 +33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 = 608. What does that have to do with the factors of 608?

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-07

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• 608 is a composite number.
• Prime factorization: 608 = 2 x 2 x 2 x 2 x 2 x 19, which can be written 608 = (2^5) x 19
• The exponents in the prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 x 2 = 12. Therefore 608 has exactly 12 factors.
• Factors of 608: 1, 2, 4, 8, 16, 19, 32, 38, 76, 152, 304, 608
• Factor pairs: 608 = 1 x 608, 2 x 304, 4 x 152, 8 x 76, 16 x 38, or 19 x 32
• Taking the factor pair with the largest square number factor, we get √608 = (√16)(√38) = 4√38 ≈ 24.657656.

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606 is the sum of the six prime numbers from 89 to 109

606 is also the hypotenuse of the Pythagorean triple 120-594-606. What is the greatest common factor of those three numbers?

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-01

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• 606 is a composite number.
• Prime factorization: 606 = 2 x 3 x 101
• 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 606 has exactly 8 factors.
• Factors of 606: 1, 2, 3, 6, 101, 202, 303, 606
• Factor pairs: 606 = 1 x 606, 2 x 303, 3 x 202, or 6 x 101
• 606 has no square factors that allow its square root to be simplified. √606 ≈ 24.617067

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605 is the hypotenuse of the Pythagorean triple 363-484-605. The greatest common factor of those three numbers should be fairly easy to spot.

6 – 0 + 5 = 11, obviously a multiple of 11, so 605 passes 11’s divisibility test and can be evenly divided by 11.

Coincidentally, OEIS.org shared some trivia about 605’s digits: 6 + 0 + 5 = 11, the greatest prime factor of 605.

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-01

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• 605 is a composite number.
• Prime factorization: 605 = 5 x 11 x 11, which can be written 605 = 5 x (11^2)
• The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1) = 2 x 3  = 6. Therefore 605 has exactly 6 factors.
• Factors of 605: 1, 5, 11, 55, 121, 605
• Factor pairs: 605 = 1 x 605, 5 x 121, or 11 x 55
• Taking the factor pair with the largest square number factor, we get √605 = (√121)(√5) = 11√5 ≈ 24.5967

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