1276 is the Third Number in a Row with Exactly 12 Factors

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1276 is the 125th number to have exactly 12 factors. I’ve made a list of those numbers in the graphic below. Two consecutive numbers appearing on the list have only happened three times before. Those numbers are highlighted in blue. 1276 is special because with it, for the first time THREE consecutive numbers appear on the list!

Look at the prime factorizations of those three consecutive numbers:
1274 = 2·7²·13
1275 = 3·5²·17
1276 = 2²·11·29

How are they the same? Can you figure out a reason why they all have exactly 12 factors?

By the way, prime number 1277 spoiled the streak especially since 1278 = 2·3²·71 and also has 12 factors!

If you came up with a rule, I think you should know that not all numbers with 12 factors will follow that same rule. For example,
2³·3² = 72 and has 12 factors because 4·3=12.
2⁵·3 = 96 and has 12 factors because 6·2 = 12.

I hope that strengthens your hypothesis instead of destroying it!

Now I’ll tell you some more facts about the number 1276:

  • 1276 is a composite number.
  • Prime factorization: 1276 = 2 × 2 × 11 × 29, which can be written 1276 = 2² × 11 × 29
  • 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 × 2 × 2 = 12. Therefore 1276 has exactly 12 factors.
  • Factors of 1276: 1, 2, 4, 11, 22, 29, 44, 58, 116, 319, 638, 1276
  • Factor pairs: 1276 = 1 × 1276, 2 × 638, 4 × 319, 11 × 116 22 × 58, or 29 × 44
  • Taking the factor pair with the largest square number factor, we get √1276 = (√4)(√319) = 2√319 ≈ 35.72114

1276 is the sum of 12 consecutive prime numbers:
79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 = 1276

1276 is the hypotenuse of a Pythagorean triple:
880-924-1276 which is (20-21-29) times 44

Finally, from OEIS.org, we learn that 1276 = 1111 + 22 + 77 + 66.

1275 is the 50th Triangular Number AND the 500th Number Whose Square Root Can Be Simplified

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I found two reasons to celebrate the number 1275: It is the 50th Triangular number, and it is the 500th number whose square root can be simplified.

First I’ll celebrate its square root by listing the 401st to the 500th numbers with simplifiable square roots. Having three or more simplifiable square roots in a row doesn’t happen that often, so I like to highlight them when it happens. 1274 and 1275 are highlighted because 1276 also has a square root that can be simplified:

If you’re wondering what are the first 400 numbers with simplifiable square roots, you can click on the graphics below that will give you 100 at a time:

1st 100 reducible square roots 2nd 100 reducible square roots Reducible Square Roots 516-765

Now to celebrate that 1275 is the 50th triangular number, I’ve arranged $12.75 in a triangle:

1275 can also be evenly divided by 5, and 25, in other words, by nickels and quarters!

Nickels won’t make a triangle but they can form a trapezoid. Here’s how I made this one: 1275 ÷ 5 = 255 which is 300 (the 24th triangular number) minus 45 (the 9th triangular number). Thus we can make $12.75 by arranging 255 nickels in a trapezoid with a top base of 10, a bottom base of 24, and a height of 15.

We can also use quarters to make a trapezoid. Here’s what I did: 1275 ÷ 25 = 51 which is 66 (the 11th triangular number) minus 15 (the 5th triangular number). Thus, $12.75 can be made by arranging 51 quarters in a trapezoid with a top base of 6, a bottom base of 11, and a height of 6.

Can you find any rectangular ways to arrange the coins to total $12.75?

Here’s a little more about the number 1275:

  • 1275 is a composite number.
  • Prime factorization: 1275 = 3 × 5 × 5 × 17, which can be written 1275 = 3 × 5² × 17
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1275 has exactly 12 factors.
  • Factors of 1275: 1, 3, 5, 15, 17, 25, 51, 75, 85, 255, 425, 1275
  • Factor pairs: 1275 = 1 × 1275, 3 × 425, 5 × 255, 15 × 85, 17 × 75, or 25 × 51,
  • Taking the factor pair with the largest square number factor, we get √1275 = (√25)(√51) = 5√51 ≈ 35.70714

1275 is the hypotenuse of SEVEN Pythagorean triples:
195-1260-1275 which is 15 times (13-84-85)
261-1248-1275 which is 3 times (87-416-425)
357-1224-1275 which is (7-24-25) times 51
540-1155-1275 which is 15 times (36-77-85)
600-1125-1275 which is (8-15-17) times 75
765-1020-1275 which is (3-4-5) times 255
891-912-1275 which is 3 times (297-304-425)

1274 Imagining With Sain Smart Jr.’s Tetris Puzzle

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My granddaughter recently received a Sain Smart Jr. Tetris Puzzle for her birthday. I got to watch as she and her younger sister had lots of fun creating pictures using the colorful pieces. Here are some of their creations:

 

As they played they experimented and learned what they liked and what they didn’t like. In the process, they learned some mathematics and may not have even realized it.

Yeah, they could also make all the Tetris puzzle pieces fit in the frame. It seems to help if you save some of the five wood grained pieces for last.

From what I could tell, this was a very fun and educational toy.

Now here I’ll share some information about the number 1274:

  • 1274 is a composite number.
  • Prime factorization: 1274 = 2 × 7 × 7 × 13, which can be written 1274 = 2 × 7² × 13
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1274 has exactly 12 factors.
  • Factors of 1274: 1, 2, 7, 13, 14, 26, 49, 91, 98, 182, 637, 1274
  • Factor pairs: 1274 = 1 × 1274, 2 × 637, 7 × 182, 13 × 98, 14 × 91, or 26 × 49,
  • Taking the factor pair with the largest square number factor, we get √1274 = (√49)(√26) = 7√26 ≈ 35.69314

35² + 7² = 1274

1274 is the hypotenuse of a Pythagorean triple:
490-1176-1274 which is (5-12-13) times 98.
It is also 2(35)( 7), 35² – 7², 35² + 7²

1273 and Level 3

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What’s the only common factor or 12 and 11? Write factors of 12 and 11 where they belong on the puzzle below. Then starting back up at the top of the puzzle, go down the puzzle writing the appropriate factors cell by cell until you’re done. You can do this!

Print the puzzles or type the solution in this excel file: 12 factors 1271-1280

Here’s some information about the number 1273:

  • 1273 is a composite number.
  • Prime factorization: 1273 = 19 × 67
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1273 has exactly 4 factors.
  • Factors of 1273: 1, 19, 67, 1273
  • Factor pairs: 1273 = 1 × 1273 or 19 × 67
  • 1273 has no square factors that allow its square root to be simplified. √1273 ≈ 35.67913

1273 is also palindrome 10011111001 in BASE 2.

 

1272 and Level 2

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Some of the factor pairs needed to solve this puzzle may be easier for you to find than others, but I’m sure you can still find all of them. Give it a try!

Print the puzzles or type the solution in this excel file: 12 factors 1271-1280

Now I’ll write a little bit about the number 1272:

  • 1272 is a composite number.
  • Prime factorization: 1272 = 2 × 2 × 2 × 3 × 53, which can be written 1272 = 2³ × 3 × 53
  • 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 × 2 × 2 = 16. Therefore 1272 has exactly 16 factors.
  • Factors of 1272: 1, 2, 3, 4, 6, 8, 12, 24, 53, 106, 159, 212, 318, 424, 636, 1272
  • Factor pairs: 1272 = 1 × 1272, 2 × 636, 3 × 424, 4 × 318, 6 × 212, 8 × 159, 12 × 106, or 24 × 53
  • Taking the factor pair with the largest square number factor, we get √1272 = (√4)(√318) = 2√318 ≈ 35.66511

1272 is the sum of four consecutive prime numbers, and it is the sum of two consecutive prime numbers:
311 + 313 + 317 + 331 = 1272
631 + 641 = 1272

1272 is the hypotenuse of a Pythagorean triple:
672-1080-1272 which is 24 times (28-45-52)

 

1271 and Level 1

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All the clues in this level 1 puzzle have a greatest common factor. If you can figure out what that GCF is, then you’ve completed the first step necessary to solve the puzzle.

Print the puzzles or type the solution in this excel file: 12 factors 1271-1280

Now I’ll write a few facts about the number 1271:

  • 1271 is a composite number.
  • Prime factorization: 1271 = 31 × 41
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1271 has exactly 4 factors.
  • Factors of 1271: 1, 31, 41, 1271
  • Factor pairs: 1271 = 1 × 1271 or 31 × 41
  • 1271 has no square factors that allow its square root to be simplified. √1271 ≈ 35.65109

1271 is the sum of the nineteen prime numbers from 29 to 107.
It is also the sum of three consecutive primes: 419 + 421 + 431 = 1271

1271 is the hypotenuse of a Pythagorean triple:
279-1240-1271 which is 31 times (9-40-41)

 

1270 What’s Brewing on My 5-Year Blogiversary

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As Halloween approaches, I remember that five years ago today, I hit the publish button for the first time, and my puzzles became available for anyone with an internet connection to use.

Today’s puzzle looks a little bit like a cauldron. What’s brewing on my 5-year blogiversary?

Print the puzzles or type the solution in this excel file: 10-factors-1259-1270

I continue to be very grateful to WordPress and the WordPress community for making blogging and publishing easy and enjoyable. I am also very grateful to my readers who have done so much to make this blog grow.

I’m a lot busier now than I was five years ago. Besides blogging, I have a full-time job and a part-time job. I like both of these jobs because I like helping students understand mathematics better. Sometimes I don’t have the time I would like to work on my blog. Nevertheless, I still have blogging goals I want to reach so lately I find myself playing catch-up more often than not.

Now I’ll write a little about the number 1270:

  • 1270 is a composite number.
  • Prime factorization: 1270 = 2 × 5 × 127
  • 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 × 2 × 2 = 8. Therefore 1270 has exactly 8 factors.
  • Factors of 1270: 1, 2, 5, 10, 127, 254, 635, 1270
  • Factor pairs: 1270 = 1 × 1270, 2 × 635, 5 × 254, or 10 × 127
  • 1270 has no square factors that allow its square root to be simplified. √1270 ≈ 35.63706

1270 is the hypotenuse of a Pythagorean triple:
762-1016-1270 which is (3-4-5) times 254

1269 Tangram Witch Puzzle

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If you have the seven tangram pieces, then you can create this Halloween witch riding across the moon (or a paper plate). Best Witches creating all kinds of things with those fabulous tiles!

In case you would like to know some facts about the number 1269, here’s what I’ve learned:

  • 1269 is a composite number.
  • Prime factorization: 1269 = 3 × 3 × 3 × 47, which can be written 1269 = 3³ × 47
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1269 has exactly 8 factors.
  • Factors of 1269: 1, 3, 9, 27, 47, 141, 423, 1269
  • Factor pairs: 1269 = 1 × 1269, 3 × 423, 9 × 141, or 27 × 47
  • Taking the factor pair with the largest square number factor, we get √1269 = (√9)(√141) = 3√141 ≈ 35.62303

1269 is the difference of two squares four different ways:
37² – 10² = 1269
75² – 66² = 1269
213² – 210² = 1269
635² – 634² = 1269

1268 Halloween Cat Mystery

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Cats can be quite mysterious. They are a favorite pet for many every day, even though suspicious stories abound about them on Halloween. Can you solve the mystery of this cat-like puzzle?

Print the puzzles or type the solution in this excel file: 10-factors-1259-1270

Now I’ll share a few facts about the number 1268:

  • 1268 is a composite number.
  • Prime factorization: 1268 = 2 × 2 × 317, which can be written 1268 = 2² × 317
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 1268 has exactly 6 factors.
  • Factors of 1268: 1, 2, 4, 317, 634, 1268
  • Factor pairs: 1268 = 1 × 1268, 2 × 634, or 4 × 317
  • Taking the factor pair with the largest square number factor, we get √1268 = (√4)(√317) = 2√317 ≈ 35.60899

28² + 22² = 1268

1268 is the hypotenuse of a Pythagorean triple:
300-1232-1268 calculated from 28² – 22², 2(28)(22), 28² + 22².
It is also 4 times (75-308-317)

1267 Frankenstein Mystery

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There are legends of Dr. Frankenstein creating a monster years ago. Nowadays Frankenstein’s Monster can often be seen walking through neighborhoods on Halloween night. This puzzle looks a little bit like him.

Print the puzzles or type the solution in this excel file: 10-factors-1259-1270

But if you take all the color away, he looks completely different and quite harmless:

Now I’ll share some information about the number 1267:

  • 1267 is a composite number.
  • Prime factorization: 1267 = 7 × 181
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1267 has exactly 4 factors.
  • Factors of 1267: 1, 7, 181, 1267
  • Factor pairs: 1267 = 1 × 1267 or 7 × 181
  • 1267 has no square factors that allow its square root to be simplified. √1267 ≈ 35.59494

1267 is the sum of nine consecutive prime numbers:
113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 + 163 = 1267

1267 is the hypotenuse of a Pythagorean triple:
133-1260-1267 which is 7 times (19-180-181)