## A Multiplication Based Logic Puzzle

### 783 and Level 2

• 783 is a composite number.
• Prime factorization: 783 = 3 x 3 x 3 x 29, which can be written 783 = (3^3) x 29
• The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 783 has exactly 8 factors.
• Factors of 783: 1, 3, 9, 27, 29, 87, 261, 783
• Factor pairs: 783 = 1 x 783, 3 x 261, 9 x 87, or 27 x 29
• Taking the factor pair with the largest square number factor, we get √783 = (√9)(√87) = 3√87 ≈ 27.982137.

Here’s today’s puzzle. It’s a level 2 so it isn’t very difficult:

Print the puzzles or type the solution on this excel file: 12-factors-782-787

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27 x 29 = 783. Since (n – 1)(n + 1) always equals n² – 1, we know that 783 is one number away from the next perfect square.

29 is a factor of 783, making 783 the hypotenuse of a Pythagorean triple:

• 540-567-783, which is 27 times 20-21-29.
• Thus 540² + 567² = 783² just as 20² + 21² = 29².

783 is also a palindrome in bases 15, 23, and 28:

• 373 BASE 15; note that 3(225) + 7(15) + 3(1) = 783
• 1B1 BASE 23 (B is 11 base 10); note that 1(23²) + 11(23) + 1(1) = 783
• RR BASE 28 (R is 27 base 10); note that 27(28) + 27 = 783

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### Reducible Square Roots up to √765

• 765 is a composite number.
• Prime factorization: 765 = 3 x 3 x 5 x 17, which can be written 765 = (3^2) x 5 x 17
• The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 765 has exactly 12 factors.
• Factors of 765: 1, 3, 5, 9, 15, 17, 45, 51, 85, 153, 255, 765
• Factor pairs: 765 = 1 x 765, 3 x 255, 5 x 153, 9 x 85, 15 x 51, or 17 x 45
• Taking the factor pair with the largest square number factor, we get √765 = (√9)(√85) = 3√85 ≈ 27.658633.

765 is the 300th number whose square root can be reduced! Here are three tables with 100 reducible square roots each showing all the reducible square roots up to √765. When three or more consecutive numbers have reducible square roots, I highlighted them.

That’s 300 reducible square roots found for the first 765 counting numbers. 300 ÷ 765 ≈ 0.392, so 39.2% of the numbers so far have reducible square roots.

Today’s puzzle is a whole lot less complicated than all that, so give it a try!

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

Logical steps to find the solution are in a table at the bottom of the post.

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Here are some other fun facts about the number 765:

765 is made from three consecutive numbers so it is divisible by 3. The middle of those numbers is 6 so 765 is also divisible by 9.

765 can be written as the sum of two squares two different ways:

• 27² + 6² = 765
• 21² + 18² = 765

Its other two prime factors, 5 and 17, have a remainder of 1 when divided by 4 so 765² can be written as the sum of two squares FOUR different ways, two of which contain other numbers that use the same digits as 765. Also notice that 9 is a factor of each number in the corresponding Pythagorean triples.

• 117² + 756² = 765²
• 324² + 693² = 765²
• 360² + 675² = 765²
• 459² + 612² = 765²

765 can also be written as the sum of three squares four different ways:

• 26² + 8² + 5² = 765
• 22² + 16² + 5² = 765
• 20² + 19² + 2² = 765
• 20² + 14² + 13² = 765

765 is a palindrome in two different bases:

• 1011111101 BASE 2; note that 1(512) + 0(256) + 1(128) + 1(64) + 1(32) + 1(16) + 1(8) + 1(4) + 0(2) + 1(1) = 765.
• 636 BASE 11; note that 6(121) + 3(11) + 6(1) = 765.

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### 456 An Inchworm Measuring Marigolds

456 is the sum of consecutive prime numbers in two different ways. One of my readers listed those ways in the comments. The factors of 456 are at the end of the post.

Inchworm, inchworm,
Measuring the marigolds
You and your arithmetic will probably go far.

Two plus two is four
Four plus four is eight
Eight and eight is sixteen
Sixteen and sixteen is thirty-two.

Inchworm, inchworm,
Measuring the marigolds
Seems to me you’d stop and see
How beautiful they are.

Today I taught a class of three year olds about being thankful for birds, insects, and creeping things. To keep their attention, I used a variety of stories, riddles, books, and games. I also sang a few songs including this one about an inchworm who is very good at arithmetic. I think preschool children can still enjoy songs like this even if they don’t understand everything the song is about or even if they are wiggling as much as an inchworm while they listen to it. Here is the song sung by Danny Kaye from the movie Hans Christian Andersen:

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Now for the number 456. The last two digits can be evenly divided by four, so the entire number is divisible by four. Also since it is formed from three consecutive numbers, it is divisible by 3. However since the number in the middle of those consecutive numbers is not 3, 6, 9 or another multiple of 3, we know that 456 is NOT divisible by 9.

Because it is divisible by four, we will use that fact first to determine how to reduce its square root.

456 ÷ 4 = 114. Notice that 114 is even, but 14 can’t be evenly divided by 4, so 114 cannot be either. Also notice that 114 is still divisible by 3. If we’re not sure whether or not 114 has any square factors, we are less likely to make a mistake if we divide it by 6 once, instead of by 2 and then by 3.

114 ÷ 6 = 19, a prime number, and we are certain there were no other square factors. Since we know 19 x 6 = 114, let’s backtrack a little and go back to that original one layer cake:

Take the square root of everything on the outside of the cake and get √456 = (√4)(√114) = 2√114

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• 456 is a composite number.
• Prime factorization: 456 = 2 x 2 x 2 x 3 x 19, which can be written 456 = (2^3) x 3 x 19
• The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 456 has exactly 16 factors.
• Factors of 456: 1, 2, 3, 4, 6, 8, 12, 19, 24, 38, 57, 76, 114, 152, 228, 456
• Factor pairs: 456 = 1 x 456, 2 x 228, 3 x 152, 4 x 114, 6 x 76, 8 x 57, 12 x 38, or 19 x 24
• Taking the factor pair with the largest square number factor, we get √456 = (√4)(√114) = 2√114 ≈ 21.3542

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Picture credits: Inchworm and ruler: http://www.kindergartenkindergarten.com/2012/06/problem-solving-measurement.html;

### Reducible Square Roots of Numbers up to 352

• 352 is even and therefore a composite number.
• Prime factorization: 352 = 2 x 2 x 2 x 2 x 2 x 11, which can be written 352 = (2^5) x 11
• The exponents in the prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 x 2 = 12. Therefore 352 has exactly 12 factors.
• Factors of 352: 1, 2, 4, 8, 11, 16, 22, 32, 44, 88, 176, 352
• Factor pairs: 352 = 1 x 352, 2 x 176, 4 x 88, 8 x 44, 11 x 32, or 16 x 22
• Taking the factor pair with the largest square number factor, we get √352 = (√16)(√22) = 4√22 ≈ 18.762

Mathematics is full of interesting patterns. Let’s explore some patterns in reducible square roots.

Here are the first ten numbers that have reducible square roots. Notice that five of the numbers in this list, 1, 4, 9, 16, and 25 are perfect squares:

40% of the numbers up to 5 have reducible square roots. The same thing is true for 40% of the numbers up to 10, 40% of the numbers up to 20, and 40% of the numbers up to 25.

Here are the second ten numbers with reducible square roots. The last number,50, is double the previous last number, 25. Again 40% of the numbers up to 50 have reducible square roots, and only two of these numbers are perfect squares. Notice that the last three numbers under the radical sign in this set are consecutive numbers.

Would you like to make any predictions for the third set of ten numbers with reducible square roots?

If you predicted that this set of numbers would end with 75, you were almost right! 29/75 or 38.67% of the numbers up to 75 have reducible square roots, and 30/76 or 39.47% of the numbers up to 76 do. Both of these values are very close to 40%, but not quite there. Notice this time only one number is a perfect square.

Would you like to make a prediction for what will happen with the fourth set of ten numbers with reducible square roots?

Surprise! We’re back to 40% of the numbers up to 100 have reducible square roots. Those consecutive numbers at the end of the set really helped raise the percentage right at the last minute. This set of numbers has two perfect squares.

What do you think will happen if we look at all the reducible square roots up to 350? Multiples of any perfect square will always have reducible square roots. Since we have more perfect squares, do you think more will be reducible?

Because the square roots of 351 and 352 are also reducible, let’s include them in this chart. Each column has 20 reducible square roots in it, and they are grouped into fives for easier counting. I’ve highlighted sets of three or four consecutive numbers. In all, we now have charts showing the first 140 reducible square roots. This last set has eight perfect squares. Let’s look at the percentages at the end of some of those sets of consecutive numbers: 50/126 or 39.68% of the first 126 numbers have reducible square roots. 96/245 or 39.18% of the first 245 numbers have reducible square roots. Finally, 140/352 or 39.77% of the numbers up to 352 have reducible square roots. That one is so close to 40%!

Sets of 41-60, 61-80, 81-100, 101-120, & 121-140 Reducible Square Roots

No matter how big a table we make, the percentage of reducible square roots will be very close to 40%. It will not get significantly higher because, believe it or not, most numbers are either prime numbers or the product of two or more DIFFERENT primes and thus have square roots that are NOT reducible.