• 778 is a composite number.
• Prime factorization: 778 = 2 x 389
• 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 778 has exactly 4 factors.
• Factors of 778: 1, 2, 389, 778
• Factor pairs: 778 = 1 x 778 or 2 x 389
• 778 has no square factors that allow its square root to be simplified. √778 ≈ 27.892651.

Here’s today’s puzzle. Hints for solving it are in a table at the end of the post.

 

Print the puzzles or type the solution on this excel file: 10-factors-2016

—————————————

OEIS.org informs us that 778 is the number of ways  a 1 x 5 rectangle can be surrounded by other 1 x 5 rectangles. I’m not going to publish or even find all 778 possible ways, but here is one of them:

27² + 7² = 778.

We can use those numbers to compute a Pythagorean triple: 2(27)(7), 27² – 7², 27² + 7², which turns out to be 378-680-778, and thus 378² + 680² = 778².

778 is also the sum of three squares three different ways:

• 25² + 12² + 3² = 778
• 24² + 11² + 9² = 778
• 21² + 16² + 9² = 778

778 is palindrome 1G1 BASE 21 (G is 16 base 10)

That palindrome means that 1(21²) + 16(21) + 1(1) = 778.

—————————————

Happy pi day to lovers of mathematics who happen to write their dates mm/dd/yyyy or mm/dd/yy (even for only one day a year)!

Lots will be written by others about pi today, but I’m going to write about the number 771 instead.

• 771 is a composite number.
• Prime factorization: 771 = 3 x 257
• 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 771 has exactly 4 factors.
• Factors of 771: 1, 3, 257, 771
• Factor pairs: 771 = 1 x 771 or 3 x 257
• 771 has no square factors that allow its square root to be simplified. √771 ≈ 27.76688675.

The prime factors of 771 are 3 and 257, both of which are Fermat primes.  Wikipedia informs us that a regular 771-sided polygon can therefore be constructed using only a straightedge and a compass. Today’s puzzle looks a little like a compass and a straightedge:

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

————————————-

Here is more about the number 771:

Like pi, √771 is irrational. The first few digits of √771 are 27.76688.

771 is the sum of three consecutive prime numbers: 251 + 257 + 263 = 771. One of those prime numbers is also a factor of 771.

Because 257 is one of its factors, 771 is the hypotenuse of a Pythagorean triple, and 96² + 765² = 771².

771 is also the sum of three squares four different ways. Notice that 11 and/or 19 appear in every one of those ways:

• 25² + 11² + 5² = 771
• 23² + 11² + 11² = 771
• 19² + 19² + 7² = 771
• 19² + 17² + 11² = 771

771 is a palindrome in FIVE different bases:

• 1100000011 BASE 2; note that 1(2^9) + 1(2^8) + 0(2^7) + 0(2^6) + 0(2^5) + 0(2^4) + 0(2^3) + 0(2^2) + 1(2^1) + 1(2^0) = 771.
• 30003 BASE 4; note that 3(4^4) + 0(4^3) + 0(4^2) + 0(4^1) + 3(4^0) = 771.
• 474 BASE 13; note that 4(169) + 7(13) + 4(1) = 771.
• 303 BASE 16; note that 3(256) + 0(16) + 3(1) = 771.
• 1D1 BASE 22 (D is 13 base 10); note that 1(22²) + 13(22) + 1(1) = 771.

————————————-

---

• 764 is a composite number.
• Prime factorization: 764 = 2 x 2 x 191, which can be written 764 = (2^2) x 191
• 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 764 has exactly 6 factors.
• Factors of 764: 1, 2, 4, 191, 382, 764
• Factor pairs: 764 = 1 x 764, 2 x 382, or 4 x 191
• Taking the factor pair with the largest square number factor, we get √764 = (√4)(√191) = 2√191 ≈ 27.6405499.

Here’s today’s puzzle:

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

—————————————

Here’s some more thoughts I had about the number 764:

Every number that ends in 4 is divisible by 2.

Every number that ends in 64 is divisible by 4.

Every number that ends in 764 is NOT divisible by 8, but every number ending in 764 can be written as the sum of 8 consecutive numbers.

For example, 92 + 93 + 94 + 95 + 96 + 97 + 98 + 99 = 764.

The first four digits of √764  are 27.64.

Wikipedia tells us that 764 is one of only two 3-digit “telephone numbers“. (911 is NOT the other one.)

—————————————

Today’s Puzzle:

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-18

Some possible steps to solve the puzzle:

Factors of 750:

• 750 is a composite number.
• Prime factorization: 750 = 2 x 3 x 5 x 5 x 5, which can be written 750 = 2 x 3 x (5^3)
• The exponents in the prime factorization are 1, 1, and 3. Adding one to each and multiplying we get (1 + 1)(1 + 1)(3 + 1) = 2 x 2 x 4 = 16. Therefore 750 has exactly 16 factors.
• Factors of 750: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 125, 150, 250, 375, 750
• Factor pairs: 750 = 1 x 750, 2 x 375, 3 x 250, 5 x 150, 6 x 125, 10 x 75, 15 x 50, or 25 x 30
• Taking the factor pair with the largest square number factor, we get √750 = (√25)(√30) = 5√30 ≈ 27.386127875.

Sum-Difference Puzzles:

30 has four factor pairs. One of those pairs adds up to 13, and another one subtracts to 13. Put the factors in the appropriate boxes in the first puzzle.

750 has eight factor pairs. One of the factor pairs adds up to ­65, and a different one subtracts to 65. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?

More reasons to be interested in the number 750:

750 can be written as the sum of consecutive numbers seven ways:

• 249 + 250 + 251 = 750; that’s 3 consecutive numbers.
• 186 + 187 + 188 + 189 = 750; that’s 4 consecutive numbers.
• 148 + 149 + 150 + 151 + 152 = 750; that’s 5 consecutive numbers.
• 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 = 750; that’s 12 consecutive numbers.
• 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 = 750; that’s 15 consecutive numbers.
• 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 = 750; that’s 20 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 + 42 = 750; that’s 25 consecutive numbers.

750 is also the sum of all the prime numbers from 17 to 79. Do you remember what those sixteen prime numbers are?

Because 5, 25, and 125 are its factors, 750 is the hypotenuse of three Pythagorean triple triangles:

• 210² + 720² = 750²
• 264² + 702² = 750²
• 450² + 600² = 750²

750² = 562500 which is another cool looking square number whose digits include 5 and the value of 5^4.

750 is also the sum of three squares six different ways:

• 26² + 7² + 5² = 750
• 25² + 11² + 2² = 750
• 25² + 10² + 5² = 750
• 23² + 14² + 5² = 750
• 23² + 11² + 10² = 750
• 19² + 17² + 10² = 750

Wikipedia tells us that 750 is the 15th nonagonal number because 15(7⋅15 – 5)/2 = 750. It is also 10 times the 5th nonogonal number because 10 ⋅ 5(7⋅5 – 5)/2 = 750.

750 is a palindrome in three different bases:

• 23232 BASE 4; note that 2(4^4) + 3(4^3) + 2(4^2) + 3(4^1) + 2(4^0) = 750.
• 2A2 BASE 17 (A= 10 base 10); note that 2(17²) + 10(17) + 2(1) = 750.
• PP BASE 29 (P = 25 base 10); note that 25(29) + 25(1) = 750.

• 742 is a composite number.
• Prime factorization: 742 = 2 x 7 x 53
• 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 742 has exactly 8 factors.
• Factors of 742: 1, 2, 7, 14, 53, 106, 371, 742
• Factor pairs: 742 = 1 x 742, 2 x 371, 7 x 106, or 14 x 53
• 742 has no square factors that allow its square root to be simplified. √742 ≈ 27.2396769.

Here is a level 4 puzzle for you to try:

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-11

————————————-

Here’s more about composite number 742:

742 can be expressed as the sum of consecutive numbers three ways:

• 184 + 185 + 186 + 187 = 742; that’s 4 consecutive numbers.
• 103 + 104 + 105 + 106 + 107 + 108 + 109 = 742; that’s 7 consecutive numbers.
• 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 + 39 + 40 = 742; that’s 28 consecutive numbers.

Because 53 is one of its prime factors, 742 is the hypotenuse of Pythagorean triple 392-630-742. The greatest common factor of three numbers in the triple is in the factor pair with 53.

742 is the sum of three squares two different ways:

• 27² + 3² + 2² = 742
• 25² + 9² + 6² = 742

————————————-

• 735 is a composite number.
• Prime factorization: 735 = 3 x 5 x 7 x 7, which can be written 735 = 3 x 5 x (7^2)
• The exponents in the prime factorization are 1, 1, and 2. Adding one to each and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 x 2 x 3 = 12. Therefore 735 has exactly 12 factors.
• Factors of 735: 1, 3, 5, 7, 15, 21, 35, 49, 105, 147, 245, 735
• Factor pairs: 735 = 1 x 735, 3 x 245, 5 x 147, 7 x 105, 15 x 49, or 21 x 35
• Taking the factor pair with the largest square number factor, we get √735 = (√49)(√15) = 7√15 ≈ 27.11088.

Here’s a medium level factoring puzzle for you to try:

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

———————————————————

735 = 7²·3·5. OEIS.org informs us that 735 is the smallest number whose digits and distinct prime factors are the same.

735 is the sum of consecutive numbers many different ways:

• 367 + 368 = 735; that’s 2 consecutive numbers.
• 244 + 245 + 246 = 735; that’s 3 consecutive numbers.
• 145 + 146 + 147 + 148 + 149 = 735; that’s 5 consecutive numbers.
• 120 + 121 + 122 + 123 + 124 + 125 = 735; that’s 6 consecutive numbers.
• 102 + 103 + 104 + 105 + 106 + 107 + 108 = 735; that’s 7 consecutive numbers.
• 69 + 70 + 71 + 72 + 73 + 74 + 75 + 76 + 77 + 78 = 735; that’s 10 consecutive numbers.
• 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 = 735; that’s 14 consecutive numbers.
• 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 = 735; that’s 15 consecutive numbers.
• 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 = 735; that’s 21 consecutive numbers.
• 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 + 39 = 735; that’s 30 consecutive numbers.
• 4  + 5 + 6 + 7 + 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 = 735; that’s 35 consecutive numbers.

Because 5 is one of its factors, 735 is the hypotenuse of Pythagorean triple 441-588-735. What is the greatest common factor of those three numbers? 147, the other half of the factor pair with 5.

735 is a palindrome in two other bases:

• 3223 BASE 6; note that 3(216) + 2(36) + 2(6) + 3(1) = 735.
• LL BASE 34 (L = 21 base 10); note that 21(24) + 21(1) = 735.

If I were calculating the square root of 735, I would make a little factor cake using only prime numbers and prime numbers squared on the outside of the cake, and then I would take the square root of everything on the outside of the cake:

 

———————————————————