• 272 is a composite number.
• Prime factorization: 272 = 2 x 2 x 2 x 2 x 17, which can be written (2^4) x 17
• 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 272 has 10 factors.
• Factors of 272: 1, 2, 4, 8, 16, 17, 34, 68, 136, 272
• Factor pairs: 272 = 1 x 272, 2 x 136, 4 x 68, 8 x 34, or 16 x 17
• Taking the factor pair with the largest square number factor, we get √272 = (√16)(√17) = 4√17 ≈ 16.492

Give this Level 4 puzzle a try!

Print the puzzles or type the factors on this excel file: 12 Factors 2014-10-20

• 264 is a composite number.
• Prime factorization: 264 = 2 x 2 x 2 x 3 x 11, which can be written (2^3) x 3 x 11
• 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 264 has 16 factors.
• Factors of 264: 1, 2, 3, 4, 6, 8, 11, 12, 22, 24, 33, 44, 66, 88, 132, 264
• Factor pairs: 264 = 1 x 264, 2 x 132, 3 x 88, 4 x 66, 6 x 44, 8 x 33, 11 x 24, or 12 x 22
• Taking the factor pair with the largest square number factor, we get √264 = (√4)(√66) = 2√66 ≈ 16.248

That’s a lot of factors. Finding the factors for this puzzle will be much easier:

Print the puzzles or type the factors on this excel file: 10 Factors 2014-10-13

• 256 is a composite number.
• Prime factorization: 256 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2, which can be written 256 = 2⁸
• The exponent in the prime factorization is 8. Adding one we get (8 + 1) = 9. Therefore 256 has 9 factors.
• Factors of 256: 1, 2, 4, 8, 16, 32, 64, 128, 256
• Factor pairs: 256 = 1 x 256, 2 x 128, 4 x 64, 8 x 32, or 16 x 16
• 256 is a perfect square. √256 = 16.

((2²)²)² = 256

1 + 3 + 5 + 7 + 9 + 11 + … + 29 + 31 = 256; that’s the sum of the first 16 odd numbers.

Because it is a power of 2, it is impossible to write 256 as the sum of consecutive whole numbers.

ALL of the factors of 256 are powers of 2: 2⁰=1, 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32, 2⁶=64, 2⁷=128, 2⁸=256

NONE of the clues in this level 4 puzzle are powers of 2, but can you still solve it and figure out where the powers of 2 belong in the puzzle?

Print the puzzles or type the factors on this excel file: 12 Factors 2014-10-06

• 249 is a composite number.
• Prime factorization: 249 = 3 x 83
• 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 249 has 4 factors.
• Factors of 249: 1, 3, 83, 249
• Factor pairs: 249 = 1 x 249 or 3 x 83
• 249 has no square factors so its square root cannot be simplified. √249 ≈ 15.7797

Print the puzzles or type the factors on this excel file: 10 Factors 2014-09-29

• 235 is a composite number.
• Prime factorization: 235 = 5 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 235 has 4 factors.
• Factors of 235: 1, 5, 47, 235
• Factor pairs: 235 = 1 x 235 or 5 x 47
• Since 235 has no square number factors, its square root cannot be simplified. √235 ≈ 15.3297

Print the puzzles or type the factors on this excel file: 10 Factors 2014-09-15

• 222 is a composite number.
• Prime factorization: 222 = 2 x 3 x 37
• 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 222 has 8 factors.
• Factors of 222: 1, 2, 3, 6, 37, 74, 111, 222
• Factor pairs: 222 = 1 x 222, 2 x 111, 3 x 74, or 6 x 37
• 222 has no square factors so its square root cannot be simplified. √222 ≈ 14.8997

Excel file of puzzles and previous week’s factor solutions: 10 Factors 2014-09-01

Today’s Puzzle:

It’s been a busy summer for me and lots of other people, yet Alan Parr found the time this summer to write a great post about the Find the Factors puzzles.  You can read it at established1962 find-the-factors.

Excel file of puzzles and previous week’s factor solutions: 12 Factors 2014-08-25

Steps That Use Logic to Solve That Puzzle:

Factors of 216:

• 216 is a composite number.
• 216 = 6 x 6 x 6, so it is a perfect cube, 6³.
• Prime factorization: 216 = 2 x 2 x 2 x 3 x 3 x 3, which can be written 216 = 2³ x 3³.
• The exponents in the prime factorization are 3 and 3. Adding one to each and multiplying we get (3 + 1) x (3 + 1) = 4 x 4 = 16. Therefore 216 has 16 factors.
• Factors of 216: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216
• Factor pairs: 216 = 1 x 216, 2 x 108, 3 x 72, 4 x 54, 6 x 36, 8 x 27, 9 x 24, 12 x 18
• Taking the factor pair with the largest square number factor, we get √216 = (√6)(√36) = 6√6 ≈ 14.697

Sum-Difference Puzzle:

54 has four factor pairs. One of those factor pairs adds up to 15, and another one subtracts to 15. Those factor pairs will help you solve the first puzzle below.

216 has eight factor pairs. One of them adds up to 30 and a different one subtracts to 30. If you can identify those factor pairs, then you can solve the second puzzle.

The second puzzle is really just the first puzzle in disguise. Why did I say that?