1206 and Level 6

If you carefully study all the clues in this Level 6 puzzle and use logic, you should be able to solve the puzzle. Stick with it and you’ll succeed!

Print the puzzles or type the solution in this excel file: 10-factors-1199-1210

Here are some facts about the number 1206:

  • 1206 is a composite number.
  • Prime factorization: 1206 = 2 × 3 × 3 × 67, which can be written 1206 = 2 × 3² × 67
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1206 has exactly 12 factors.
  • Factors of 1206: 1, 2, 3, 6, 9, 18, 67, 134, 201, 402, 603, 1206
  • Factor pairs: 1206 = 1 × 1206, 2 × 603, 3 × 402, 6 × 201, 9 × 134, or 18 × 67,
  • Taking the factor pair with the largest square number factor, we get √1206 = (√9)(√134) = 3√134 ≈ 34.72751

Notice that 6·201 = 1206. Not very many numbers can equal themselves by using their own digits in a different way with +, -, ×, ÷, and/or parenthesis. That fact makes 1206 only the seventeenth Friedman Number.

1206 is also the sum of the twenty prime numbers from 19 to 103.

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1022 Friedman Number Mystery

1022 is the 15th Friedman number. “What is a Friedman number and why is 1022 one of them?” you may ask. I will solve that little mystery for you. 1022 is a Friedman number because
2¹⁰ – 2 = 1022. Notice that the expression 2¹⁰ – 2 uses the digits 1, 0, 2, and 2 in some order and a subtraction sign. A Friedman number can be written as an expression that uses all of its own digits the exact number of times that they occur in the number. The expression must include at least one operator (+, -, ×, ÷) or a power. Parenthesis are allowed as long as the other rules are followed.

Now I would like you to solve the mystery of this puzzle using logic and the multiplication facts. Can you do it?

Print the puzzles or type the solution in this excel file: 10-factors-1019-1027

1022 is the hypotenuse of a Pythagorean triple:
672-770-1022 which is 14 times (48-55-73)

  • 1022 is a composite number.
  • Prime factorization: 1022 = 2 × 7 × 73
  • 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 × 2 × 2 = 8. Therefore 1022 has exactly 8 factors.
  • Factors of 1022: 1, 2, 7, 14, 73, 146, 511, 1022
  • Factor pairs: 1022 = 1 × 1022, 2 × 511, 7 × 146, or 14 × 73
  • 1022 has no square factors that allow its square root to be simplified. √1022 ≈ 31.96873

688 is a Friedman number

688 is a palindrome in two different bases:

  • 2002 in base 7; note that 2(343) + 0(49) + 0(7) + 2(1) = 688
  • 494 in base 12; note that 4(144) + 9(12) + 4(1) = 688

688 is called a Friedman number because it can be expressed using only its own digits and +, -, x, ÷, ( ), and exponents. Stetson.edu has published a table of all such numbers that have 4 digits or less and the reason each qualifies to be a Friedman number.

688 is a Friedman number because 688 = 8 x 86 so I made a factor tree based on that single multiplication fact:

Since it’s such a fun number fact, I positioned it on top of today’s factoring puzzle, too.

688 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

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  • 688 is a composite number.
  • Prime factorization: 688 = 2 x 2 x 2 x 2 x 43, which can be written 688 = 2⁴ x 43
  • 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 688 has exactly 10 factors.
  • Factors of 688: 1, 2, 4, 8, 16, 43, 86, 172, 344, 688
  • Factor pairs: 688 = 1 x 688, 2 x 344, 4 x 172, 8 x 86, or 16 x 43
  • Taking the factor pair with the largest square number factor, we get √688 = (√16)(√43) = 4√43 ≈ 26.229754.

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688 Factors