## A Multiplication Based Logic Puzzle

### 623 and Level 2

623 is the hypotenuse of Pythagorean triple 273-560-623. What is the greatest common factor of those three numbers?

623 is not divisible by 2, 3, or 5. Is 623 divisible by 7? You can apply either of the following divisibility rules after you separate 623’s digits into 62 and 3:

1st rule: 62 – 2(3) = 62 – 6 = 56. Since 56 is divisible by 7, 623 is divisible by 7.

2nd rule: 62 + 5(3) = 62 + 15 = 77. Since 77 is divisible by 7, 623 is divisible by 7.

86 + 87 + 88 + 89 + 90 + 91 + 92 = 623  (7 consecutive numbers because 623 is divisible by 7)

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

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• 623 is a composite number.
• Prime factorization: 623 = 7 x 89
• 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 623 has exactly 4 factors.
• Factors of 623: 1, 7, 89, 623
• Factor pairs: 623 = 1 x 623 or 7 x 89
• 623 has no square factors that allow its square root to be simplified. √623 ≈ 24.9599679.

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### A Teacher Affects Eternity and Divisibility Tricks Applied to 418

Henry Brooks Adams said, “A teacher affects eternity; he can never tell where his influence stops.” Of course the quote applies to female teachers just as much as it does to male teachers. Affecting eternity is pretty serious business.

It will probably take less than two minutes to read Marek Bennet’s  comic on how the work you do fits into the educational system. Its message is worth pondering for quite a long time.

Let me demonstrate some quick prime number divisibility tricks on the number 418:

• It’s even so its divisible by 2.
• 4 + 1 + 8 = 13, which is not a multiple of 3, so 418 is not divisible by 3.
• The last digit isn’t 0 or 5, so 418 is not divisible by 5.
• 41 – 2(8) = 41 – 16 = 25, which is not a multiple of 7, so 418 is not divisible by 7.
• If we add the first and the 3rd digits of 418, we get twelve. Then if we subtract the second digit, one, we get eleven. Since we got a multiple of eleven, that means that our original number, 418, can be evenly divided by eleven.

Here is the factoring information for 418:

• 418 is a composite number.
• Prime factorization: 418 = 2 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 418 has exactly 8 factors.
• Factors of 418: 1, 2, 11, 19, 22, 38, 209, 418
• Factor pairs: 418 = 1 x 418, 2 x 209, 11 x 38, or 19 x 22
• 418 has no square factors that allow its square root to be simplified. √418 ≈ 20.445

### 398 and Level 4

Let’s apply a couple of divisibility rules to the number 398 to find some of its factors. 2 is a factor of 398 because 8 is even. 4 is NOT a factor of 398 because 4 is NOT a factor of 98. We know this because 8 is divisible by 4, and 9 is an odd number.

Print the puzzles or type the factors on this excel file: 10 Factors 2015-02-16

• 398 is a composite number.
• Prime factorization: 398 = 2 x 199
• 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 398 has exactly 4 factors.
• Factors of 398: 1, 2, 199, 398
• Factor pairs: 398 = 1 x 398 or 2 x 199
• 398 has no square factors that allow its square root to be simplified. √398 ≈ 19.9499

### 120 and Level 5

120  is a composite number. 120 = 1 x 120, 2 x 60, 3 x 40, 4 x 30, 5 x 24, 6 x 20, 8 x 15, or 10 x 12. Factors of 120: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120. Prime factorization: 120 = 2 x 2 x 2 x 3 x 5, which can also be written 120 = 2³ x 3 x 5.

Thinking process using divisibility tricks to find the factor pairs of 120:

√120 is irrational and approximately equal to 10.95. Every factor pair of 120 will have one factor less than 10.95 and one factor greater than 10.95, and we will find both factors in each pair at the same time. The following numbers are less than 10.95. Are they factors of 120?

1. Yes, all whole numbers are divisible by 1, so 1 x 120 = 120.
2. Yes, 120 is an even number. 120 ÷ 2 = 60, so 2 x 60 = 120. (Since 60 is even, 4 will also be a factor of 120.)
3. Yes, 1 + 2 + 0 = 3 which is divisible by 3 (but not by 9), so 120 is divisible by 3. 120 ÷ 3 = 40, so 3 x 40 = 120. Note 120 will not be divisible by 9.
4. Yes, the number formed from its last two digits, 20, is divisible by 4, so 120 is divisible by 4, and 4 x 30 = 120. (Note since 30 is even, 8 will also be a factor of 120.)
5. Yes, the last digit is 0 or 5, so 120 is divisible by 5, and 5 x 24 = 120.
6. Yes, 120 is divisible by both 2 and 3, so it is divisible by 6, and 6 x 20 = 120.
7. No. The divisibility trick for 7 requires us to split 120 into 12 and 0. We double 0 and subtract the double from 12. 12 – (2 x 0) = 12 – 0 = 12. Since 12 is not divisible by 7, 120 also is not divisible by 7.
8. Yes, see 4 above. 120 = 8 x 15. (This will mean that ANY number whose last 3 digits are 120 will also be divisible by 8.)
9. No, see 3 above. 120 is not divisible by 9.
10. Yes, 120 ends with a zero, so 10 is a factor of 120, and 10 x 12 = 120.

From this thinking process we conclude that the factor pairs of 120 are 1 x 120, 2 x 60, 3 x 40, 4 x 30, 5 x 24, 6 x 20, 8 x 15, and 10 x 12.

When 120 is a clue in the FIND THE FACTORS 1 – 12 puzzles, use 10 and 12 as the factors.

5! = 1 x 2 x 3 x 4 x 5 = 120.

Excel file of this week’s puzzles and last week’s factors: 10 Factors 2014-05-12