• 282 is a composite number.
• Prime factorization: 282 = 2 x 3 x 47
• 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 282 has 8 factors.
• Factors of 282: 1, 2, 3, 6, 47, 94, 141, 282
• Factor pairs: 282 = 1 x 282, 2 x 141, 3 x 94, or 6 x 47
• 282 has no square factors that allow its square root to be simplified. √282 ≈ 16.793

Today is Halloween. Give this Level 5 puzzle a try. . . Trick or Treat?

Print the puzzles or type the factors on this excel file: 10 Factors 2014-10-27

• 280 is a composite number.
• Prime factorization: 280 = 2 x 2 x 2 x 5 x 7, which can be written 280 = 2³ x 5 x 7
• 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 280 has 16 factors.
• Factors of 280: 1, 2, 4, 5, 7, 8, 10, 14, 20, 28, 35, 40, 56, 70, 140, 280
• Factor pairs: 280 = 1 x 280, 2 x 140, 4 x 70, 5 x 56, 7 x 40, 8 x 35, 10 x 28, or 14 x 20
• Taking the factor pair with the largest square number factor, we get √280 = (√4)(√70) = 2√70 ≈ 16.733

When I was a child, my brother showed me how he could multiply using his fingers. The same method is clearly illustrated here. If you haven’t seen it before, take a look. It isn’t the fastest method for multiplying, but it is still fun to see that it works. You can also practice the multiplication facts by finding the factors that solve this puzzle and then completing the multiplication table.

Print the puzzles or type the factors on this excel file: 10 Factors 2014-10-27

 

• 278 is a composite number.
• Prime factorization: 278 = 2 x 139
• 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 278 has 4 factors.
• Factors of 278: 1, 2, 139, 278
• Factor pairs: 278 = 1 x 278 or 2 x 139
• 278 has no square factors that allow its square root to be simplified. √278 ≈ 16.673

Here is today’s puzzle:

Print the puzzles or type the factors on this excel file: 10 Factors 2014-10-27

A Logical Approach to FIND THE FACTORS: 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.

• 276 is a composite number.
• Prime factorization: 276 = 2 x 2 x 3 x 23, which can be written 276 = (2^2) x 3 x 23
• 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 276 has 12 factors.
• Factors of 276: 1, 2, 3, 4, 6, 12, 23, 46, 69, 92, 138, 276
• Factor pairs: 276 = 1 x 276, 2 x 138, 3 x 92, 4 x 69, 6 x 46, or 12 x 23
• Taking the factor pair with the largest square number factor, we get √276 = (√4)(√69) = 2√69 ≈ 16.613

I’m excited that resourceaholic included a FIND THE FACTORS puzzle in a gems post. Check it out! There are several other interesting resources there, too.

Print the puzzles or type the factors on this excel file: 10 Factors 2014-10-27

• 274 is a composite number.
• Prime factorization: 274 = 2 x 137
• 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 274 has 4 factors.
• Factors of 274: 1, 2, 137, 274
• Factor pairs: 274 = 1 x 274 or 2 x 137
• 274 has no square factors that allow its square root to be simplified. √274 ≈ 16.553.

I recommend using logic and not guess and check to solve the Find the Factors puzzles, but the logic for the one below can be a little tricky. If necessary, scroll down for one piece of reasoning that may be helpful if you get stuck in the middle of trying to solve this one.

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

Notes on some logic needed for this puzzle:

• The choices for 24 are 2 x 12, 3 x 8, or 4 x 6.
• Because 28 = 4 x 7, 16 = 2 x 8, not 4 x 4. Thus 16 will use one of the 2’s. Either 6 or 8 will use the other 2, so we eliminate  2 x 12 = 24.
• Also 21 will use one of the 3’s while 6 or 18 will use the other. Thus we eliminate  3 x 8 = 24.
• Therefore 24 = 4 x 6 or 6 x 4.

 

• 273 is a composite number.
• Prime factorization: 273 = 3 × 7 × 13
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2= 8. Therefore 273 has exactly 8 factors.
• Factors of 273: 1, 3, 7, 13, 21, 39, 91, 273
• Factor pairs: 273 = 1 × 273, 3 × 91, 7 × 39, or 13 × 21
• 273 has no square factors that allow its square root to be simplified. √273 ≈ 16.523

Find the Factors is now one of  many  interesting mathematical resources that can be found on Solvemymaths!

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

• 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

Today’s Puzzle:

Can you solve this level 2 puzzle?

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

Factors of 270:

• 270 is a composite number.
• Prime factorization: 270 = 2 x 3 x 3 x 3 x 5, which can be written 2 x (3^3) x 5
• The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 x 4 x 2 = 16. Therefore 270 has 16 factors.
• Factors of 270: 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, 135, 270
• Factor pairs: 270 = 1 x 270, 2 x 135, 3 x 90, 5 x 54, 6 x 45, 9 x 30, 10 x 27, or 15 x 18
• Taking the factor pair with the largest square number factor, we get √270 = (√9)(√30) = 3√30 ≈ 16.432

Sum-Difference Puzzles:

30 has four factor pairs. One of those factor pairs adds up to 13, and another subtracts to 13. Can you find those factor pairs to solve the first puzzle below?

270 has eight factor pairs. One of them adds up to 39, and a different one subtracts to 39. 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 would I say that?

Factors for Today’s Puzzle: