• 772 is a composite number.
• Prime factorization: 772 = 2 x 2 x 193, which can be written 772 = (2^2) x 193
• 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 772 has exactly 6 factors.
• Factors of 772: 1, 2, 4, 193, 386, 772
• Factor pairs: 772 = 1 x 772, 2 x 386, or 4 x 193
• Taking the factor pair with the largest square number factor, we get √772 = (√4)(√193) = 2√193 ≈ 27.78488798.

Here is a factoring puzzle to try:

Print the puzzles or type the solution on this excel file: 12 Factors 2016-02-25

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Here’s more about the number 772:

24² + 14² = 772 so 772 is the hypotenuse of a Pythagorean triple, and 380² + 672² = 772². 380-672-772 is calculated from 24² – 14², 2(24)(14) , 24² + 14².

22² + 12² + 12² = 772, making 772 the sum of three square numbers.

772 is also the sum of two consecutive prime numbers: 383 + 379 = 772.

OEIS.org informs us that 772 is the smallest number that is the sum of three triangular numbers 21 different ways. I decided to find all those ways for myself and share them here. (If zero wasn’t named the zeroth triangular number, there would “only” be 20 ways.)

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• 736 is a composite number.
• Prime factorization: 736 = 2 x 2 x 2 x 2 x 2 x 23, which can be written 732 = (2^5) x 23
• 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 736 has exactly 12 factors.
• Factors of 736: 1, 2, 4, 8, 16, 23, 32, 46, 92, 184, 368, 736
• Factor pairs: 736 = 1 x 736, 2 x 368, 4 x 184, 8 x 92, 16 x 46, or 23 x 32
• Taking the factor pair with the largest square number factor, we get √736 = (√16)(√46) = 4√46 ≈ 27.1293199.

Some great online resources for teachers are on the Mathfireworks website. I am very pleased that Find the Factors made the list. It is included under Puzzles and Games. Help for solving today’s puzzle can be found in the table at the bottom of this post.

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-04

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23 ⋅ 32 = 736 so it is the product of semordnilaps. (Semordnilap is palindromes spelled backwards, so semordnilap and palindromes are semordnilaps.)

736 = 7 + 3^6 making it the 14^th Friedman number. Since 736 is equal to an expression that uses only “+ – × / ^ ( )”, all of its digits (or a concatenation of its digits) in the order in which they occur in the number, it is the 3rd nice Friedman number. Thank you OEIS.org for that fun number fact. (All of the suggested links are different and well worth a look.) Note: (736) = 736 is trivial so it would not satisfy the definition of a Friedman number.

21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 = 736; that’s 23 consecutive numbers.

735 and 736 are the smallest pair of consecutive numbers with 12 factors each.

736 is a palindrome in 2 bases:

• 1E1 BASE 21 (E = 14 base 10); note that 1(21^2) + 14(21^1) + 1(21^0) = 736.
• NN BASE 31 (N = 23 base 10); note that 23(31) + 23(1) = 736.

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Two years have 730 days when neither year is a leap year.

• 730 is a composite number.
• Prime factorization: 730 = 2 x 5 x 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 x 2 x 2 = 8. Therefore 730 has exactly 8 factors.
• Factors of 730: 1, 2, 5, 10, 73, 146, 365, 730
• Factor pairs: 730 = 1 x 730, 2 x 365, 5 x 146, or 10 x 73
• 730 has no square factors that allow its square root to be simplified. √730 ≈ 27.01851.

Here are some statistics that WordPress gave me just before 2015 ended: findthefactors.com 2015annual report.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-28

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Here a few more facts about the number 730:

730 is the sum of consecutive numbers several ways:

• 181 + 182 + 183 + 184 = 730; that’s 4 consecutive numbers.
• 144 + 145 + 146 + 147 + 148 = 730; that’s 5 consecutive numbers.
• 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 = 730; that’s 20 consecutive numbers.

730 is also the sum of two squares two ways:

• 27² + 1² = 730
• 21² + 17² = 730

Because 5 and 73 are two of its factors, 730 is the hypotenuse of FOUR Pythagorean triples (none are primitive):

• 54-728-730, calculated from 2(27)(1), 27² – 1², 27² + 1²
• 152-714-730, calculated from 21² – 17², 2(21)(17), 21² + 17²
• 438-584-730, and 146 is their GCF (greatest common factor).
• 480-550-730, and 10 is their GCF.

730 is a palindrome in three bases:

• 1000001 BASE 3
• 1001 BASE 9
• 101 BASE 27

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• 722 is a composite number.
• Prime factorization: 722 = 2 x 19 x 19, which can be written 722 = 2 x (19^2)
• The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1) = 2 x 3  = 6. Therefore 722 has exactly 6 factors.
• Factors of 722: 1, 2, 19, 38, 361, 722
• Factor pairs: 722 = 1 x 722, 2 x 361, or 19 x 38
• Taking the factor pair with the largest square number factor, we get √722 = (√361)(√2) = 19√2 ≈ 26.870058.

Factoring 722 is a lot easier if you’ve memorized that 19² = 361.

Here is today’s factoring puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2015-12-21

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Here are some other interesting facts about the number 722:

I learned from OEIS.org that (2^4) + (3^4) + (5^4) = 722. The red numbers are the first 3 prime numbers.

722 is also the sum of consecutive numbers two different ways:

• 179 + 180 + 181 + 182 = 722; that’s 4 consecutive numbers.
• 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39+ 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 = 722; that’s 19 consecutive numbers.

722 is a palindrome in two other bases:

• 2D2 BASE 16 (D = 13 base 10); note that 2(16²) + 13(16) + 2(1) = 722.
• 242 BASE 18; note that 2(18²) + 4(18) + 2(1) = 722.

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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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