1092 Predictable Factor Trees

A couple of years ago on Memorial Day weekend, we bought two peach trees from a local nursery. Those trees have grown bigger, and we will get some peaches this year. The smaller tree has lots of fruit growing on it and will need some attention because the branches will be too small to support the weight of all that fruit. The bigger tree has exactly one peach growing on it.

These factor trees for the number 1092 are a lot more predictable than those peach trees: No matter which of its factor pairs you use, you will always eventually get
2² × 3 × 7 × 13 = 1092.

Here are some more facts about 1092:

  • 1092 is a composite number.
  • Prime factorization: 1092 = 2 × 2 × 3 × 7 × 13, which can be written 1092 = 2² × 3 × 7 × 13
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1092 has exactly 24 factors.
  • Factors of 1092: 1, 2, 3, 4, 6, 7, 12, 13, 14, 21, 26, 28, 39, 42, 52, 78, 84, 91, 156, 182, 273, 364, 546, 1092
  • Factor pairs: 1092 = 1 × 1092, 2 × 546, 3 × 364, 4 × 273, 6 × 182, 7 × 156, 12 × 91, 13 × 84, 14 × 78, 21 × 52, 26 × 42, or 28 × 39
  • Taking the factor pair with the largest square number factor, we get √1092 = (√4)(√273) = 2√273 ≈ 33.04542

(12 × 13 × 14)/2 = 1092
Even though 1092 can claim that cool fact, it will actually make the next number a STAR!

1092 is the sum of the twelve prime numbers from 67 to 113:
67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 = 1092

1092 has a lot of factors, but it is the hypotenuse of only one Pythagorean triple:
420-1008-1092 which is (5-12-13) times 84

Look how cool 1092 is in some other bases:

Not only does it use only 0’s and 1’s in base 2 but also in consecutive bases 3 and 4:
It’s 10001000100 in BASE 2 because 2¹º + 2⁶ + 2² = 1092,
1111110 in BASE 3 because 3⁶ + 3⁵ + 3⁴ + 3³ + 3² + 3¹ = 1092,
and 101010 in BASE 4 because  4⁵ + 4³ + 4¹ = 1092

I like the way it looks in consecutive bases 12 and 13:
It’s 770 in BASE 12, because 7(12² + 12) = 7(156) = 1092
and 660 in BASE 13 because 6(13² + 13) = 6(182) = 1092

And its repdigit 444 in BASE 16 because 4(16² + 16 + 1) = 1092

Some of these facts about 1092 were predictable and some were not, but I have enjoyed learning all of them and hope that you have too.

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What’s Special About √1024?

What’s special about √1024? Is it because it and several counting numbers after it have square roots that can be simplified?

Perhaps.

Maybe it is interesting just because √1024 = 32, a whole number. The 5th root of 1024 = 4 and the 10th root of 1024 = 2, both whole numbers as well.

Those equations are true because 32² = 1024, 4⁵ = 1024, and 2¹⁰ = 1024.

Or perhaps 1024 is special because it is the smallest number that is a 10th power. (It is 2¹⁰.) The square root of a perfect 10th power is always a perfect 5th power. (32 = 2⁵ and is the smallest number that is a 5th power.)

1024 is also the smallest number with exactly 11 factors.

It is the smallest number whose factor tree has at least 10 leaves that are prime numbers. (They are the red leaves on the factor tree shown below.) It is possible to draw several other factor trees for 1024, but they will all have the number 2 appearing ten times.

What’s more, I noticed something about 1024 and some other multiples of 256: Where do multiples of 256 fall on the list of square roots that can be simplified?

  • 256 × 1 = 256 and 256 is the 100th number on this list of numbers whose square roots can be simplified.

1st 100 reducible square roots

  • 256 × 2 = 512. When we add the next 100 square roots that can be simplified, 512 is the 199th number on the list.

2nd 100 reducible square roots

  • Here are the third 100 square roots that can be simplified:

Reducible Square Roots 516-765

  • 256 × 3 = 768 didn’t quite make that list because it is the 301st number. Indeed, it is the first number on this list of the fourth 100 numbers whose square roots can be simplified.

  • 256 × 4 = 1024. That will be the first number on the 5th 100 square roots list!

It is interesting that those multiples of 256 have the 100th, the 199th, the 301st, and the 401st positions on the list. That is so close to the 100th, 200th, 300th, and 400th positions.

In case you couldn’t figure it out, the highlighted square roots are three or more consecutive numbers that appear on the list.

1024 is interesting for many other reasons. Here are a few of them:

(4-2)¹⁰ = 1024, making 1024 the 16th Friedman number.

I like to remember that 2¹⁰ = 1024, which is just a little bit more than a thousand. Likewise 2²⁰ = 1,048,576 which is about a million. 2³⁰ is about a billion, and 2⁴⁰ is about a trillion.

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As stated in the comments, Paula Beardell Krieg shared a related post with me. It takes exactly 1024 Legos to build this fabulous pyramidal fractal:

https://platform.twitter.com/widgets.js
1024 has so many factors that are divisible by 4 that it is a leg in NINE Pythagorean triples:
768-1024-1280 which is (3-4-5) times 256
1024-1920-2176 which is (8-15-17) times 128
1024-4032-4160 which is (16-63-65) times 64
1024-8160-8224 which is (32-255-257) times 32
1024-16368-16400 which is 16 times (64-1023-1025)
1024-32760-32776 which is 8 times (128-4095-4097)
1024-65532-65540 which is 4 times (256-16383-16385)
1024-131070-131074 which is 2 times (512-65535-65537),
and primitive 1024-262143-262145

Some of those triples can also be found because 1024 is the difference of two squares four different ways:
257² – 255² = 1024
130² – 126² = 1024
68² – 60² = 1024
40² – 24² = 1024
To find out which difference of two squares go with which triples, add the squares instead of subtracting and you’ll get the hypotenuse of the triple.
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1024 looks interesting in some other bases:
It’s 1000000000 in BASE 2,
100000 in BASE 4,
2000 in BASE 8,
1357 in BASE 9,
484 in BASE 15,
400 in BASE 16,
169 in BASE 29,
144 in BASE 30,
121 in BASE 31, and
100 in BASE 32

  • 1024 is a composite number.
  • Prime factorization: 1024 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2, which can be written 1024 = 2¹⁰
  • The exponent in the prime factorization is 10. Adding one we get (10 + 1) = 11. Therefore 1024 has exactly 11 factors.
  • Factors of 1024: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
  • Factor pairs: 1024 = 1 × 1024, 2 × 512, 4 × 256, 8 × 128, 16 × 64, or 32 × 32,
  • 1024 is a perfect square. √1024 = 32. It is also a perfect 5th power, and a perfect 10th power.

1020 A Week of Mystery

Sometimes changing things up a little is good. I decided to make a week’s worth of mystery level puzzles. The actual difficulty level will vary from puzzle to puzzle so give each one of them a try. If you think one is too easy or too difficult, the next one might not be. Here’s the first one:

Print the puzzles or type the solution in this excel file: 10-factors-1019-1027

Here’s a little bit about the number 1020:

It is only 8 more than 1008, the previous number that also had 24 factors. Here are a few of its MANY possible factor trees.

1020 is the sum of six consecutive prime numbers:
157 + 163 + 167 + 173 + 179 + 181 = 1020

1020 is the hypotenuse of four Pythagorean triples:
156-1008-1020 which is 12 times (13-84-85)
432-924-1020 which is 12 times (36-77-85)
480-900-1020 which is (8-15-17) times 60
612-816-1020 which is (3-4-5) times 204

1020 looks interesting when it is written using some different bases:
It’s 33330 in BASE 4 because 3(4⁴ + 4³ + 4² + 4¹) = 3(340) = 1020,
848 in BASE 11 because 8(11²) + 4(11) + 8(1) = 1020,
606 in BASE 13 because 6(13²) + 6(1) = 6(170) = 1020,
480 in BASE 15 because 4(15²) + 8(15) = 4(225 + 30) = 4(255) = 1020,
390 in BASE 17 because 3(17²) + 9(17) = 3(289 + 51) = 3(340) = 1020,
UU in BASE 33 (U is 30 base 10) because 30(33) + 30(1) = 30(34) = 1020, and
U0 in BASE 34 because 30(34) = 1020

  • 1020 is a composite number.
  • Prime factorization: 1020 = 2 × 2 × 3 × 5 × 17, which can be written 1020 = 2² × 3 × 5 × 17
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1020 has exactly 24 factors.
  • Factors of 1020: 1, 2, 3, 4, 5, 6, 10, 12, 15, 17, 20, 30, 34, 51, 60, 68, 85, 102, 170, 204, 255, 340, 510, 1020
  • Factor pairs: 1020 = 1 × 1020, 2 × 510, 3 × 340, 4 × 255, 5 × 204, 6 × 170, 10 × 102, 12 × 85, 15 × 68, 17 × 60, 20 × 51, or 30 × 34
  • Taking the factor pair with the largest square number factor, we get √1020 = (√4)(√255) = 2√255 ≈ 31.9374

A Forest of 1008 Factor Trees

The number 1008 has so many factors that I just had to make a forest of some of its MANY possible factor trees. 1008 has fifteen factor pairs. I’ve made a factor tree for every factor pair, except 1 × 1008. I’ll start off with these short, wide, beautiful trees that feature six of 1008’s factor pairs:

Notice that no matter what factor pair we use, each tree has the same prime factors that I have highlighted in red. Even this lean, thin very-easy-to-read tree uses those same prime factors:

Finally, here are eight more factor trees that begin with the other eight factor pairs for 1008. They aren’t as good-looking as all the trees above, but they still work as factor trees and help us find those same red prime factors for 1008:

Here are some other facts about the number 1008:

1008 is the sum of ten consecutive prime numbers:
79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 = 1008

1008 looks interesting when written in some other bases:
It’s 33300 in BASE 4 because 3(4⁴) + 3(4³) + 3(4²) = 3(256 + 64 + 16) = 3(336) = 1008,
4400 in BASE 6 because 4(6³) + 4(6²) = 4(216 + 36) = 4(252) = 1008,
700 in BASE 12 because 7(12²) = 7(144) = 1008
SS in BASE 35 (S is 28 base 10) because 28(35) + 28(1) = 28(36) = 1008
S0 in BASE 36 because 28(36) = 1008

  • 1008 is a composite number.
  • Prime factorization: 1008 = 2 × 2 × 2 × 2 × 3 × 3 × 7, which can be written 1008 = 2⁴ × 3² × 7
  • The exponents in the prime factorization are 4, 2 and 1. Adding one to each and multiplying we get (4 + 1)(2 + 1)(1 + 1) = 5 × 3 × 2 = 30. Therefore 1008 has exactly 30 factors.
  • Factors of 1008: 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 36, 42, 48, 56, 63, 72, 84, 112, 126, 144, 168, 252, 336, 504, 1008
  • Factor pairs: 1008 = 1 × 1008, 2 × 504, 3 × 336, 4 × 252, 6 × 168, 7 × 144, 8 × 126, 9 × 112, 12 × 84, 14 × 72, 16 × 63, 18 × 56, 21 × 48, 24 × 42, or 28 × 36
  • Taking the factor pair with the largest square number factor, we get √1008 = (√144)(√7) = 12√7 ≈ 31.74902

992 Christmas Factor Tree

Artificial Christmas trees have to be assembled. Sometimes the assembly is easy, and sometimes it is frustrating.

This Christmas tree puzzle can be solved using LOGIC and an ordinary multiplication table, but there’s a good chance it will frustrate you. Go ahead and try to solve it!

Print the puzzles or type the solution in this excel file: 10-factors-986-992

The number 992 also can make a nice looking, well-balanced factor tree:

992 is the product of two consecutive numbers: 31 × 32 = 992.
Because of that fact, 992 is the sum of the first 31 EVEN numbers:
2 + 4 + 6 + 8 + 10 + . . . + 54 + 56 + 58 + 60 + 62 = 992

992 is palindrome 212 in BASE 22 because 2(22²) + 1(22) + 2(1) = 922. That was a lot of 2’s and 1’s in that fun fact!

  • 992 is a composite number.
  • Prime factorization: 992 = 2 × 2 × 2 × 2 × 2 × 31, which can be written 732 = 2⁵ × 31
  • The exponents in the prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 992 has exactly 12 factors.
  • Factors of 992: 1, 2, 4, 8, 16, 31, 32, 62, 124, 248, 496, 992
  • Factor pairs: 992 = 1 × 992, 2 × 496, 4 × 248, 8 × 124, 16 × 62, or 31 × 32
  • Taking the factor pair with the largest square number factor, we get √992 = (√16)(√62) = 4√62 ≈ 31.49603

990 Christmas Factor Trees

Today’s puzzle has a couple of small Christmas trees in it. Don’t let their smallness fool you into thinking this is an easy puzzle. Can you solve it? Remember to use logic and not guess and check to find the solution.

Print the puzzles or type the solution in this excel file: 10-factors-986-992

There are many interesting facts about the number 990:

9 × 10 × 11 = 990

Because 44 × 45/2 = 990, it is the 44th triangular number. That means that the sum of all the numbers from 1 to 44 is 990.

990 is the sum of the twelve prime numbers from 59 to 107.
It is also the sum of six consecutive prime numbers:
151 + 157 + 163 + 167 + 173 + 179 = 990,
and the sum of two consecutive primes:
491 + 499 = 990

990 is the hypotenuse of a Pythagorean triple:
594-792-990 which is (3-4-5) times 198

990 looks interesting in some other bases:
It is 6A6 in BASE 12 (A is 10 base 10) because 6(144) + 10(12) + 6(1) = 990,
2E2 in BASE 19 (E is 14 base 10) because 2(19²) + 14(19) + 2(1) = 990
1K1 in BASE 23 (K is 20 base 10) because 1(23²) + 20(23) + 1(1) = 990
UU in BASE 32 (U is 30 base 10) because 30(32) + 30(1) = 30(33) = 990
U0 in BASE 33 because 30(33) = 990

  • 990 is a composite number.
  • Prime factorization: 990 = 2 × 3 × 3 × 5 × 11, which can be written 990 = 2 × 3² × 5 × 11
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 990 has exactly 24 factors.
  • Factors of 990: 1, 2, 3, 5, 6, 9, 10, 11, 15, 18, 22, 30, 33, 45, 55, 66, 90, 99, 110, 165, 198, 330, 495, 990
  • Factor pairs: 990 = 1 × 990, 2 × 495, 3 × 330, 5 × 198, 6 × 165, 9 × 110, 10 × 99, 11 × 90, 15 × 66, 18 × 55, 22 × 45, or 30 × 33
  • Taking the factor pair with the largest square number factor, we get √990 = (√9)(√110) = 3√110 ≈ 31.464265

980 Christmas Factor Trees

This level 4 puzzle has 12 clues in it that are products of factor pairs in which both factors are numbers from 1 to 12. The clues make an evergreen tree, the symbol of everlasting life which is so fitting for Christmas. Can you find the factors for the given clues and put them in the right places?

Print the puzzles or type the solution in this excel file: 12 factors 978-985

Now I’ll tell you a little about the number 980:

It has eighteen factors and many possible factor trees. Here are just three of them:

28² + 14² = 980, so 980 is the hypotenuse of a Pythagorean triple:
588-784-980 which is (3-4-5) times 196, but can also be calculated from
28² – 14², 2(28)(14), 28² + 14²

I like the way 980 looks in some other bases:
It is 5A5 in BASE 13 (A is 10 base 10) because 5(13) + 10(13) + 5(1) = 980,
500 in BASE 14 because 5(14²) = 980,
SS in BASE 34 (S is 28 base 10) because 28(34) + 28(1) = 28(35) = 980
S0 in BASE 35 because 28(35) = 980

  • 980 is a composite number.
  • Prime factorization: 980 = 2 × 2 × 5 × 7 × 7, which can be written 980 = 2² × 5 × 7²
  • The exponents in the prime factorization are 2, 1 and 2. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(2 + 1) = 3 × 2 × 3 = 18. Therefore 980 has exactly 18 factors.
  • Factors of 980: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 49, 70, 98, 140, 196, 245, 490, 980
  • Factor pairs: 980 = 1 × 980, 2 × 490, 4 × 245, 5 × 196, 7 × 140, 10 × 98, 14 × 70, 20 × 49 or 28 × 35
  • Taking the factor pair with the largest square number factor, we get √980 = (√196)(√5) = 14√5 ≈ 31.30495.

960 Factor Trees

960 is the smallest number to have exactly 28 factors. 960 is 2⁶·3·5, so any factor tree made for it will have 6 + 1 + 1 = 8 prime factors. Since 8 is a power of 2, this number, 960, has some beautiful and well-balanced factor trees as well as some that aren’t as good-looking. Here are five of the MANY possible factor trees for 960:

 

960 can be written as the difference of 2 squares TEN different ways:

  1. 241² – 239² = (241 + 239)(241 – 239) = 480 × 2 = 960
  2. 122² – 118² = (122 + 118)(122 – 118) = 240 × 4 = 960
  3. 83² – 77² = (83 + 77)(83 – 77) = 160 × 6 = 960
  4. 64² – 56² = (64 + 56)(64 – 56) = 120 × 8 = 960
  5. 53² – 43² = (53 + 43)(53 – 43) = 96 × 10 = 960
  6. 46² – 34² = (46 + 34)(46 – 34) = 80 × 12 = 960
  7. 38² – 22² = (38 + 22)(38 – 22) = 60 × 16 = 960
  8. 34² – 14² = (34 + 14)(34 – 14) = 48 × 20 = 960
  9. 32² – 8² = (32 + 8)(32 – 8) = 40 × 24 = 960
  10. 31² – 1² = (31 + 1)(31 – 1) = 32 × 30 = 960

960 is the sum of the sixteen prime numbers from 29 to 97.

It is also the sum of six consecutive prime numbers:
149 + 151 + 157 + 163 + 167 + 173 = 960

960 is the hypotenuse of a Pythagorean triple:
576-768-960 which is (3-4-5) times 192

I like how 960 looks in these other bases:
33000 in BASE 4 because 3(4⁴) + 3(4³) = 3(256 + 64) = 3 × 320 = 960
440 in BASE 15 because 4(15²) + 4(15) = 4(225 + 15) = 4 × 240 = 960
UU in BASE 31 (U is 30 base 10), because 30(31) + 30(1) = 30(31 + 1) = 30 × 32 = 960
U0 in BASE 32 because 30(32) + 0 = 960

Stetson.edu informs us that 9 + 6 + 09³ + 6³ + 0³ = 960

  • 960 is a composite number and a perfect square.
  • Prime factorization: 960 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 5, which can be written 960 = 2⁶ × 3 × 5
  • The exponents in the prime factorization are 6, 1 and 1. Adding one to each and multiplying we get (6 + 1)(1 + 1)(1 + 1) = 7 × 2 × 2 = 28. Therefore 960 has exactly 28 factors.
  • Factors of 960: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 64, 80, 96, 120, 160, 192, 240, 320, 480, 960
  • Factor pairs: 960 = 1 × 960, 2 × 480, 3 × 320, 4 × 240, 5 × 192, 6 × 160, 8 × 120, 10 × 96, 12 × 80, 15 × 64, 16 × 60, 20 × 48, 24 × 40, or 30 × 32
  • Taking the factor pair with the largest square number factor, we get √960 = (√64)(√15) = 8√15 ≈ 30.9838668

864 Factor Trees

The prime factorization of 864 is 2⁵ × 3³. The sum of the exponents is 5 + 3 = 8. Since 8 is a power of 2,  a couple of 864’s factor trees are full and well-balanced:

All of those prime factors lined up in numerical order. That didn’t happen for the next one, but it still makes a good looking tree, and all the prime factors are easy to find.

Is it possible to make a factor tree for 864 that hardly looks like a tree and isn’t as easy to find all the prime factors? Yes, it is. Here’s an example:

864 looked interesting to me in a few other bases:

  • 4000 BASE 6 because 4(6³) = 864
  • 600 BASE 12 because 6(12²) = 864
  • RR BASE 31 (R is 27 base 10) because 27(31) + 27(1) = 27(32) = 864
  • OO BASE 35 (O is 24 base 10) because 24(35) + 24(1) = 24(36) = 864
  • O0 BASE 36 (Oh zero) because 24(36) + 0(1) = 864

864 is the sum of the 20 prime numbers from 7 to 83.

131 + 137 + 139 + 149 + 151 + 157 = 864; that’s six consecutive primes.

431 + 433 = 864; that’s the sum of twin primes.

  • 864 is a composite number.
  • Prime factorization: 864 = 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3, which can be written 864 = 2⁵ × 3³
  • The exponents in the prime factorization are 5 and 2. Adding one to each and multiplying we get (5 + 1)(3 + 1) = 6 × 4 = 24. Therefore 864 has exactly 24 factors.
  • Factors of 864: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 72, 96, 108, 144, 216, 288, 432, 864
  • Factor pairs: 864 = 1 × 864, 2 × 432, 3 × 288, 4 × 216, 6 × 144, 8 × 108, 9 × 96, 12 × 72, 16 × 54, 18 × 48, 24 × 36, or 27 × 32
  • Taking the factor pair with the largest square number factor, we get √864 = (√144)(√6) = 12√6 ≈ 29.3938769

 

756 and Level 3

  • 756 is a composite number.
  • Prime factorization: 756 = 2 x 2 x 3 x 3 x 3 x 7, which can be written 756 = (2^2) x (3^3) x 7
  • The exponents in the prime factorization are 2, 3 and 1. Adding one to each and multiplying we get (2 + 1)(3 + 1)(1 + 1) = 3 x 4 x 2 = 24. Therefore 756 has exactly 24 factors.
  • Factors of 756: 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 27, 28, 36, 42, 54, 63, 84, 108, 126, 189, 252, 378, 756
  • Factor pairs: 756 = 1 x 756, 2 x 378, 3 x 252, 4 x 189, 6 x 126, 7 x 108, 9 x 84, 12 x 63, 14 x 54, 18 x 42, 21 x 36 or 27 x 28
  • Taking the factor pair with the largest square number factor, we get √756 = (√21)(√36) = 6√21 ≈ 27.495454.

756-factor-pairs

 

756 has many factors and, therefore, it has many possible factor trees. Here are three of them:

756 factor trees

Here’s a level 3 Find the Factors puzzle for you to solve, too:

756 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-25

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Here’s a few more thoughts on the number 756:

The last two digits of 756 is divisible by 4 so 756 is divisible by 4.

756 is formed from 3 consecutive numbers (5, 6, 7) so it is divisible by 3. The middle number is divisible by 3 so 756 is also divisible by 9.

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

  • 251 + 252 + 253 = 756; that’s 3 consecutive numbers.
  • 105 + 106 + 107 + 108 + 109 + 110 + 111 = 756; that’s 7 consecutive numbers.
  • 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 = 756; that’s 8 consecutive numbers.
  • 80 + 81 + 82 + 83 + 84 + 85 + 86 + 87 + 88 = 756; that’s 9 consecutive numbers.
  • 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 = 756; that’s 21 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 + 43 = 756; that’s 24 consecutive numbers.
  • 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 + 41  = 756; that’s 27 consecutive numbers.

756 is also the sum of six consecutive prime numbers: 109 + 113 + 127+ 131 + 137+ 139 = 756.

756 can be written as the sum of three squares four different ways. (Notice that all of the squares are even):

  • 26² + 8² + 4² = 756
  • 24² + 12² + 6² = 756
  • 22² + 16² + 4² = 756
  • 20² + 16² + 10² = 756

756 is a palindrome in two other bases:

  • 11011 BASE 5; note that 1(625) + 1(125) + 0(25) + 1(5) + 1(1) = 756.
  • LL BASE 35 (L is 21 base 10); note that 21(35) + 21(1) = 756.

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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.

756 Factors