Friedman

Why Is 1792 a Friedman Number?

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Today’s Puzzle:

I’ve mentioned before that putting a 12 in one of the last two boxes will let you avoid negative numbers as you explore the relative relationship of the clues. For this puzzle, I would suggest that you put the 12 in the third from the last box. Why? Because the last triangle on the bottom has an 8 in it, and we will need to use either 12 – 8 = 4, and 4 – 2 = 2 for the last three boxes or 11 – 8 = 3, and 3 – 2 = 1.

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After you make your way to the empty triangle on the left of the puzzle, you will notice that you are missing the numbers 1 and 8. There isn’t any way to get a 5 by subtracting those two numbers, but if you realize that 13 – 5 = 8, you should know what adjustments you need to make to solve the puzzle.

Factors of 1792:

If the last digit of a number is 2 or 6, and the next-to-the-last digit is odd, then the whole number is divisible by 4.

If the last digit of a number is 0, 4, or 8, and the next-to-the-last digit is even, then the whole number is also divisible by 4.

1792 will allow us to apply those two divisibility observations several times as we make this factor tree:

  • 1792 is a composite number.
  • Prime factorization: 1792 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 7, which can be written 1792 = 2⁸ × 7.
  • 1792 has at least one exponent greater than 1 in its prime factorization so √1792 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1792 = (√256)(√7) = 16√7.
  • The exponents in the prime factorization are 8 and 1. Adding one to each exponent and multiplying we get (8 + 1)(1 + 1) = 9 × 2 = 18. Therefore 1792 has exactly 18 factors.
  • The factors of 1792 are outlined with their factor pair partners in the graphic below.

More About the Number 1792:

1792 is a Friedman number because 7·2⁹⁻¹ = 1792.

Notice that the digits 1, 7, 9, and 2 and only those digits are used on both sides of the equal sign, and they are used the same number of times. 1792 is only the 26th Friedman number.

1792 is the difference of two squares in SEVEN different ways:
449² – 447² = 1792,
226² – 222² = 1792,
116² – 108² = 1792,
71² – 57² = 1792,
64² – 48² = 1792,
46² – 18² = 1792, and
44² – 12² = 1792.

1530 Jack-o’-lantern

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Today’s Puzzle:

Here’s a Jack-O’-Lantern Puzzle for you to enjoy. It’s a Level 5 puzzle so it might be more of a trick than a treat. Remember to use logic every step of the way instead of guessing and checking.

Here’s the same puzzle without any added color:

Factors of 1530:

15 is half of 30, so 1530 is divisible by 6 just like all these numbers are divisible by 6: 12, 24, 36, 48, 510, 612, 714, 816, 918, 1020, 1122, 1224, 1326, 1428, and so forth.

  • 1530 is a composite number.
  • Prime factorization: 1530 = 2 × 3 × 3 × 5 × 17, which can be written 1530 = 2 × 3² × 5 × 17.
  • 1530 has at least one exponent greater than 1 in its prime factorization so √1530 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1530 = (√9)(√170) = 3√170.
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1530 has exactly 24 factors.
  • The factors of 1530 are outlined with their factor pair partners in the graphic below.

More about the Number 1530:

51 × 30 = 1530. Did you notice that the same digits appear on both sides of the equal sign and only +, -, ×, ÷, (), or exponents were used to make a true statement? 1530 is only the 25th number that can make that claim, so we call it the 25th Friedman number.

There are MANY possible factor trees for 1530, but let’s celebrate that it is also a Friedman number with this one:

1530 is the hypotenuse of FOUR Pythagorean triples:
234-1512-1530, which is 18 times (13-84-85),
648-1386-1530, which is 18 times (36-77-85),
720-1350-1530, which is (8-15-17) times 90, and
918-1224-1530, which is (3-4-5) times 306.