## A Multiplication Based Logic Puzzle

### Level 2 and Simplifying √1014

Have you memorized a basic multiplication table? If you have, then you can solve this puzzle. The numbers being multiplied together aren’t where they are in a regular multiplication table, but you can still easily figure out where they need to go. There is only one solution. I bet you can find it!

Print the puzzles or type the solution in this excel file: 12 factors 1012-1018

If I wanted to find √1014, I would first check to see if it were divisible by 4 or by 9 because most numbers whose square roots can be simplified are divisible by 4 or by 9 or both.
1014 isn’t divisible by 4 because 14 isn’t divisible by 4.
It isn’t divisible by 9 because 1 + 0 + 1 + 4 = 6, and 6 is not divisible by 9.
However, it is divisible by both 2 and 3 and thus also by 6. Since most people are less likely to make a mistake dividing by 6 in ONE step instead of two, I would make a little division cake and do that division first:

Recognizing that 169 is a perfect square, I would then take the square root of everything on the outside of my little cake. (√6)(√169) = 13√6

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Here are some facts about the number 1014:

Because 13² is one of its factors, 1014 is the hypotenuse of two Pythagorean triples:
714-720-1014 which is 6 times (119-120-169),
390-936-1014 which is (5-12-13) times 78

1014 looks interesting when written in some other bases:
It’s 600 in BASE 13 because 6(13²) = 6 (169) = 1014,
and 222 in BASE 22 because 2(22²) + 2(22) + 2(1) = 2(484 + 22 + 1) = 2(507) = 1014

• 1014 is a composite number.
• Prime factorization: 1014 = 2 × 3 × 13 × 13, which can be written 1014 = 2 × 3 × 13²
• The exponents in the prime factorization are 1, 1, and 2. Adding one to each and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 × 2 × 3 = 12. Therefore 1014 has exactly 12 factors.
• Factors of 1014: 1, 2, 3, 6, 13, 26, 39, 78, 169, 338, 507, 1014
• Factor pairs: 1014 = 1 × 1014, 2 × 507, 3 × 338, 6 × 169, 13 × 78, or 26 × 39,
• Taking the factor pair with the largest square number factor, we get √1014 = (√169)(√6) = 13√6 ≈ 31.84337