Level 6 Puzzle

127 and Level 6

/ ivasallay / 2 Comments
  • 127 is a prime number.
  • Prime factorization: 127 is prime.
  • The exponent of prime number 127 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 127 has exactly 2 factors.
  • Factors of 127: 1, 127
  • Factor pairs: 127 = 1 x 127
  • 127 has no square factors that allow its square root to be simplified. √127 ≈ 11.2694

How do we know that 127 is a prime number? If 127 were not a prime number, then it would be divisible by at least one prime number less than or equal to √127 ≈ 11.3. Since 127 cannot be divided evenly by 2, 3, 5, 7, or 11, we know that 127 is a prime number.

Thinking process using divisibility tricks to prove that 127 is not divisible by 2, 3, 5, 7, or 11:

  • 127 is not even, so 127 is not divisible by 2.
  • 1 + 2 + 7 = 10 which is not divisible by 3, so 127 is not divisible by 3.
  • The last digit of 127 is not 5 or 0, so 127 is not divisible by 5.
  • The divisibility trick for 7 requires us to split 127 into 12 and 7. We double 7 and find the difference between the double and 12. We get (2 x 7) – 12  = 14 – 12 = 2. Since 2 is not divisible by 7, 127 is also not divisible by 7.
  • To perform the divisibility trick for 11 we look at the digits another way: Digits that are in an odd position are in regular type and the digit in an even position is in bold type: 127. We find the sum of the numbers typed in regular type (1 + 7  = 8) and the sum of the number(s) typed in bold type (2) and subtract the smaller sum from the larger sum (8 – 2 = 6). If the difference were a multiple of 11, than the original number would also be a  multiple of 11 (and divisible by 11). Since 6 is not a multiple of 11, we know 127 is not divisible by 11.

Therefore the only factor pair that exists for 127 is 1 x 127, and it is a prime number.

127 is never a clue in the FIND THE FACTORS puzzles.

Spotting the logic needed to begin this Level 6 puzzle should be easier than usual. If you have completed Level 5 puzzles, but have never before attempted a Level 6 puzzle, try this one !

2014-20 Level 6

Excel file of this week’s puzzles and last week’s factors: 12 Factors 2014-05-19

2014-20 Level 6 Logic

121 and Level 6

/ ivasallay / 1 Comment

121 is not a clue in today’s puzzle. However, when 121 is a clue in a FIND THE FACTORS 1 – 12 puzzle, place 11 in the corresponding cells in both the factor row and the factor column.

2014-19 Level 6

This week’s puzzles and last week’s factors: 10 Factors 2014-05-12

Here’s more about the number 121:

  • 121 is a composite number.
  • Prime factorization: 121 = 11 × 11 which can be written 121 = 11²
  • The exponent in the prime factorization is 2. Adding one we get (2 + 1) = 3. Therefore 121 has exactly 3 factors.
  • Factors of 121: 1, 11, 121
  • Factor pairs: 121 = 1 × 121 or 11 × 11
  • 121 is a perfect square. √121 = 11

121 = 11². Those digits 1, 1, 2 make 121 the second Friedman number because they can be arranged as a mathematical expression that equals a number that uses those same digits.

121 is not only a perfect square but it is also a star! In fact, games of Chinese checkers will be based on that perfect star.

Since 121 is the square of 11, the fifth prime number, it is only the fifth number to have exactly 3 factors.

121 is the sum of the three prime numbers from 37 to 43.

121 looks interesting when it is written in some other bases:
It’s repdigit 11111 in BASE 3 because 3⁴ + 3³ + 3² + 3¹ + 3⁰ = 121
and it’s 321 in BASE 6.

Not only is it a palindrome in base 10, but
it’s 232 in BASE 7, and
it’s 171 in BASE 8

It can also look pretty square depending on the base used:
It’s 441 in BASE 5,
144 in BASE 9,
121 in BASE 10 (of course),
100 in BASE 11,
81 in BASE 15, and
49 in BASE 28

121 is a pretty remarkable number!

If you got stuck solving the puzzle above, here’s one logical way to find the solution:

2014-19 Level 6 Logic

115 and Level 6

/ ivasallay / Leave a comment

115  is a composite number. 115 = 1 x 115 or 5 x 23. Factors of 115: 1, 5, 23, 115. Prime factorization: 115 = 5 x 23.

115 is never a clue in the FIND THE FACTORS puzzles.

2014-18 Level 6

Excel file of puzzles and the previous week’s solutions: 12 Factors 2014-05-05

2014-18 Level 6 Logic

107 and Level 6

/ ivasallay / Leave a comment
  • 107 is a prime number.
  • Prime factorization: 107 is prime.
  • The exponent of prime number 107 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 107 has exactly 2 factors.
  • Factors of 107: 1, 107
  • Factor pairs: 107 = 1 x 107
  • 107 has no square factors that allow its square root to be simplified. √107 ≈ 10.34408

How do we know that 107 is a prime number? If 107 were not a prime number, then it would be divisible by at least one prime number less than or equal to √107 ≈ 10.3 Since 107 cannot be divided evenly by 2, 3, 5, or 7, we know that 107 is a prime number.

107 is never a clue in the FIND THE FACTORS puzzles.

2014--17 Level 6

This week’s puzzles and last week’s solutions: 10 Factors 2014-04-28

2014-17 Level 6 Logic

 

101 and Level 6

/ ivasallay / 3 Comments
  • 101 is a prime number.
  • Prime factorization: 101 is prime.
  • The exponent of prime number 101 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 101 has exactly 2 factors.
  • Factors of 101: 1, 101
  • Factor pairs: 101 = 1 x 101
  • 101 has no square factors that allow its square root to be simplified. √101 ≈ 10.0498756

How do we know that 101 is a prime number? If 101 were not a prime number, then it would be divisible by at least one prime number less than or equal to √101 ≈ 10. Since 101 cannot be divided evenly by 2, 3, 5, or 7, we know that 101 is a prime number.

101 is never a clue in the FIND THE FACTORS puzzles.

2014-16 Level 6

This week’s puzzles and last week’s solutions: 12 Factors 2014-04-21

2014-16 Level 6 Logic

88 and April Showers

/ ivasallay / Leave a comment

88 is a composite number. 88 = 1 x 88, 2 x 44, 4 x 22, and 8 x 11. Factors of 88: 1, 2, 4, 8, 11, 22, 44, 88. Prime factorization: 88 = 2 x 2 x 2 x 11 which can also be written 88 = 2³ x 11.

Sometimes 88 is a clue in the FIND THE FACTORS 1 – 12 puzzles. Even though it has other factors, we only use 88 = 8 x 11 in the puzzles.

2014-14 Level 6

This week’s puzzles and last week’s solutions: 12 Factors 2014-04-07

Getting started logically on this puzzle is a little complicated, and I may have abbreviated too much in the chart below, so here is an explanation in regular English: The clue 14 has to use one of the 2s, so the clue 4 cannot be 2 x 2. Since 4 will be 4 x 1, and thus using one of the 4s, 16 cannot be 4 x 4, but must be 2 x 8. Since 14 and 16 use both of the 2s, 10 cannot be 2 x 5 and must be 1 x 10. Finally that means that the only clue on the puzzle that has 5 as the factor is 35, so the blank space must belong to 5 as well.

2014-14 Level 6 Logic

83 and Level 1

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2014-14 Level 1

This week’s puzzles and last week’s solutions: 12 Factors 2014-04-07

  • 83 is a prime number.
  • Prime factorization: 83 is prime.
  • The exponent of prime number 83 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 83 has exactly 2 factors.
  • Factors of 83: 1, 83
  • Factor pairs: 83 = 1 x 83
  • 83 has no square factors that allow its square root to be simplified. √83 ≈ 9.11043

How do we know that 83 is a prime number? If 83 were not a prime number, then it would be divisible by at least one prime number less than or equal to √83 ≈ 9.1. Since 83 cannot be divided evenly by 2, 3, 5, or 7, we know that 83 is a prime number.

83 is never a clue in the FIND THE FACTORS puzzles.

 2014-14 Level 1 Factors

82 and Level 6

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2014-13 Level 6

This week’s puzzles and last week’s solutions: 10 Factors 2014-03-31

82 is a composite number. 82 = 1 x 82 or  2 x 41. Factors of 82: 1, 2, 41, 82. Prime factorization: 82 = 2 x 41.

82 is never a clue in the FIND THE FACTORS puzzles.

2014-13 Level 6 Logic

75 and Level 6

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2014-12 Level 6

This week’s puzzles for you to solve

75 is a composite number. 75 = 1 x 75, 3 x 25, or 5 x 15. Factors of 75: 1, 3, 5, 15, 25, 75. Prime factorization: 75 = 3 x 5 x 5, which can also be written 75 = 3 x 5².

75 is never a clue in the FIND THE FACTORS puzzles.

75 is in this wonderful pattern. Can you tell what the pattern is?

2014-12 Level 6 Logic

68 and Happy Saint Patrick’s Day!

/ ivasallay / 1 Comment

68 is a composite number. 68 = 1 x 68, 2 x 34, or 4 x 17. Factors of 68: 1, 2, 4, 17, 34, 68. Prime factorization: 68 = 2 x 2 x 17, which can also be written 2² x 17.

68 is never a clue in the FIND THE FACTORS puzzles.

2014-11 Level 6

All Previously Published Puzzles

I survived my daughter’s wedding. The last of our out-of-state visitors will be going home tomorrow. AND my computer that broke and spent a week in the shop is working again. I will publish six new puzzles next Monday.

2014-11 Level 6 Logic