• 713 is a composite number.
• Prime factorization: 713 = 23 x 31
• 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 713 has exactly 4 factors.
• Factors of 713: 1, 23, 31, 713
• Factor pairs: 713 = 1 x 713 or 23 x 31
• 713 has no square factors that allow its square root to be simplified. √713 ≈ 26.7020598.

Go ahead and give this Level 4 puzzle a try:

 

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-14

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Would you like to know a little more about 713?

713 can be written as the sum of consecutive numbers several ways:

• 356 + 357 = 713; that’s two consecutive numbers.
• 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 = 713; that’s 23 consecutive numbers.
• 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 = 713; that’s 31 consecutive numbers.

713 is also the sum of three consecutive prime numbers: 233 + 239 + 241 = 713.

Two of its factors are 4 numbers away from their average, 27, so (23)(31) = (27 – 4)(27 + 4) = 27² – 4² = 713

713 is palindrome NN in BASE 30 (N = 23 in base 10); note that 23(30) +23(1) = 713.

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Thank you, Ricardo, for tweeting the puzzle and your work:

@findthefactors amazing for friday pic.twitter.com/JSROzz93uN

— riccardo (@ricca9380) December 18, 2015

//platform.twitter.com/widgets.js

• 712 is a composite number.
• Prime factorization: 712 = 2 x 2 x 2 x 89, which can be written 712 = (2^3) x 89
• 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 712 has exactly 8 factors.
• Factors of 712: 1, 2, 4, 8, 89, 178, 356, 712
• Factor pairs: 712 = 1 x 712, 2 x 356, 4 x 178, or 8 x 89
• Taking the factor pair with the largest square number factor, we get √712 = (√4)(√178) = 2√178 ≈ 26.683328.

Here’s a Level 3 puzzle for you to solve:

 

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-14

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Here are a few more facts about the number 712:

37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 = 712; that’s 16 consecutive numbers.

712 is the sum of the twenty-one prime numbers from 2 to 73.

It is also the sum of consecutive prime numbers 353 and 359.

Because 89 is one of its prime factors, 712 is the hypotenuse of the Pythagorean triple 312-640-712. What is the greatest common factor of those three numbers?

712 is palindrome 4B4 in BASE 12 because 4(144) + 11(12) + 4(1) = 712.

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. (None of the factors are greater than 12.)  Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

• 711 is a composite number.
• Prime factorization: 711 = 3 x 3 x 79, which can be written 711 = (3^2) x 79
• The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 711 has exactly 6 factors.
• Factors of 711: 1, 3, 9, 79, 237, 711
• Factor pairs: 711 = 1 x 711, 3 x 237, or 9 x 79
• Taking the factor pair with the largest square number factor, we get √711 = (√9)(√79) = 3√79 ≈ 26.66458.

Here is today’s puzzle:

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-14

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Even though in English 711 is a rather fun rhyming number, it has been difficult for me to find interesting facts about it:

7 + 1 + 1 = 9 so 711 is divisible by both 3 and 9.

It can be written as the sum of consecutive numbers three ways:

• 355 + 356 = 711; that’s two consecutive numbers.
• 236 + 237 + 238 = 711; that’s three consecutive numbers.
• 75 + 76 + 77 + 78 + 79 + 80 + 81 + 82 + 83 = 711; that’s nine consecutive numbers.

Is 711 a palindrome in any other base? Not in any base less than 78, but in BASE 78 it is palindrome 99 because 9(78) + 9(1) = 711.

We could ALMOST write that 711 is 11 7 in BASE 64 because 11(64) + 7(1) = 711. However, we can’t REALLY write that because base 10’s eleven would most likely be represented by the letter B in base 64. Thus we would likely write 711 as B7 in Base 64.

Since the words “eleven” and “twelve” do not contain any reference to the word “ten”, I suppose in English we could say that 711 is eleven 7 in base 64. We likely wouldn’t be able to make that same claim in other languages, however.

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• 708 is a composite number.
• Prime factorization: 708 = 2 x 2 x 3 x 59, which can be written 708 = (2^2) x 3 x 59
• 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 708 has exactly 12 factors.
• Factors of 708: 1, 2, 3, 4, 6, 12, 59, 118, 177, 236, 354, 708
• Factor pairs: 708 = 1 x 708, 2 x 354, 3 x 236, 4 x 177, 6 x 118, or 12 x 59
• Taking the factor pair with the largest square number factor, we get √708 = (√4)(√177) = 2√177 ≈ 26.608269.

Here is a factor tree for 708:

Today’s factoring puzzle reminds me of lumps of coal:

Print the puzzles or type the solution on this excel file: 10 Factors 2015-12-07

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Here are a few more facts about 708:

708 can be written as the sum of consecutive numbers three different ways:

• 235 + 236 + 237 = 708; that’s 3 consecutive numbers.
• 85 + 86 + 87 + 88 + 89 + 90 + 91 + 92 = 708; that’s 8 consecutive numbers.
• 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 = 708; that’s 24 consecutive numbers.

708 is palindrome 323 in BASE 15; note that 3(225) + 2(15) + 3(1) = 708.

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• 707 is a composite number.
• Prime factorization: 707 = 7 x 101
• 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 707 has exactly 4 factors.
• Factors of 707: 1, 7, 101, 707
• Factor pairs: 707 = 1 x 707 or 7 x 101
• 707 has no square factors that allow its square root to be simplified. √707 ≈ 26.58947.

Here is today’s factoring puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2015-12-07

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Here are a few more facts about 707:

707 is a palindrome in base 10 and in base 100.

707 is the sum of seven consecutive numbers: 98 + 99 + 100 + 101 + 102 + 103 + 104 = 707

It is also the sum of two consecutive numbers: 353 + 354 = 707.

707 is the sum of these five consecutive prime numbers: 131 + 137 + 139 + 149 + 151 = 707.

And it is the sum of the nineteen prime numbers from 5 to 73.

Since 101 is one of its prime factors, 707 is the hypotenuse of Pythagorean triple 140-693-707. What is the greatest common factor of those three numbers?

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• 706 is a composite number.
• Prime factorization: 706 = 2 x 353
• 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 706 has exactly 4 factors.
• Factors of 706: 1, 2, 353, 706
• Factor pairs: 706 = 1 x 706 or 2 x 353
• 706 has no square factors that allow its square root to be simplified. √706 ≈ 26.5706

Here is today’s puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2015-12-07

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Here are a few more facts about 706:

Because 353 is one of its prime factors, 706 is the hypotenuse of Pythagorean triple 450-544-706. What is the greatest common factor of those three numbers?

Not only is 706 the product of two prime palindromes (2 and 353), but 706 is also a palindrome in 2 different bases:

• 424 BASE 13; note 4(169) + 2(13) + 4(1) = 706
• 2C2 BASE 16; note 2(256) + 12(16) + 2(1) = 706

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. (None of the factors are greater than 10.)  Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.