Factors

1356 Mystery

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The clues that appear in this puzzle are all you need to find all the factors. Seriously. Some mysteries are easier to solve than others. Give this one a try!

Print the puzzles or type the solution in this excel file: 10 Factors 1347-1356

Here is some information about the puzzle number, 1356:

  • 1356 is a composite number.
  • Prime factorization: 1356 = 2 × 2 × 3 × 113, which can be written 1356 = 2² × 3 × 113
  • 1356 has at least one exponent greater than 1 in its prime factorization so √1356 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1356 = (√4)(√339) = 2√339
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1356 has exactly 12 factors.
  • The factors of 1356 are outlined with their factor pair partners in the graphic below.

1356 is the hypotenuse of a Pythagorean triple:
180-1344-1356 which is 12 times (15-112-113)

1355 I Can Solve This Puzzle, Can You?

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Level 6 puzzles are a little tricky because there is more than one possible common factor for every set of clues on the same row or column.

You can still solve it using logic. I can solve it. Can you?

Print the puzzles or type the solution in this excel file: 10 Factors 1347-1356

If you need some logical hints, the video below will be helpful:

Now I’ll share some information about the puzzle number, 1355:

  • 1355 is a composite number.
  • Prime factorization: 1355 = 5 × 271
  • 1355 has no exponents greater than 1 in its prime factorization, so √1355 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1355 has exactly 4 factors.
  • The factors of 1355 are outlined with their factor pair partners in the graphic below.

1355 is the hypotenuse of a Pythagorean triple:
813-1084-1355 which is (3-4-5) times 271

 

1354 Solving a Level 5 Puzzle

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What are the common factors of 16 and 4? Don’t guess which one to use. Use logic to figure it out as you find all the factors for this puzzle!

Print the puzzles or type the solution in this excel file: 10 Factors 1347-1356

If you get stuck, you can watch this video:

Now I’ll share some information about the puzzle number, 1354:

  • 1354 is a composite number.
  • Prime factorization: 1354 = 2 × 677
  • 1354 has no exponents greater than 1 in its prime factorization, so √1354 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1354 has exactly 4 factors.
  • The factors of 1354 are outlined with their factor pair partners in the graphic below.

1354 is the sum of two squares:
27² + 25² = 1354

1354 is the hypotenuse of a Pythagorean triple:
104-1350-1354 which is 2 times (52-675-677)
and can also be calculated from 27² – 25², 2(27)(25), 27² + 25²

1353 How to Solve a Level 4 Puzzle

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Try solving this level 4 puzzle. If you need help with it, I explain the steps in the video below the puzzle:

Print the puzzles or type the solution in this excel file: 10 Factors 1347-1356

Now I’ll tell you a little bit about the puzzle number, 1353:

  • 1353 is a composite number.
  • Prime factorization: 1353 = 3 × 11 × 41
  • 1353 has no exponents greater than 1 in its prime factorization, so √1353 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1353 has exactly 8 factors.
  • The factors of 1353 are outlined with their factor pair partners in the graphic below.

1353 is the hypotenuse of a Pythagorean triple:
297-1320-1353 which is 33 times (9-40-41)

1352 and Level 3

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If you know what numbers divide evenly into both 3 and 15, then you can solve this puzzle. Only use factor pairs where both numbers are from 1 to 10. Start at the top of the puzzle and work down cell by cell until you have found all the factors!

Print the puzzles or type the solution in this excel file: 10 Factors 1347-1356

Here I explain how to solve the above puzzle:

Here are a few facts about the puzzle number, 1352:

  • 1352 is a composite number.
  • Prime factorization: 1352 = 2 × 2 × 2 × 11 × 11, which can be written 1352 = 2³ × 13²
  • 1352 has at least one exponent greater than 1 in its prime factorization so √1352 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1352 = (√676)(√2) = 26√2
  • The exponents in the prime factorization are 3 and 2. Adding one to each exponent and multiplying we get (3 + 1)(2 + 1) = 4 × 3 = 12. Therefore 1352 has exactly 12 factors.
  • The factors of 1352 are outlined with their factor pair partners in the graphic below.

1352 is the hypotenuse of two Pythagorean triples:
520-1248-1352 which is (5-12-13) times 104
952-960-1352 which is 8 times (119-120-169)

 

1351 and a Level 2 Smiley Face

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You will smile a great big smile once you solve this puzzle!

Print the puzzles or type the solution in this excel file: 10 Factors 1347-1356

If you want a little help solving the puzzle, I explain how to do it in this video:

Here are some facts about the puzzle number 1351:

  • 1351 is a composite number.
  • Prime factorization: 1351 = 7 × 193
  • 1351 has no exponents greater than 1 in its prime factorization, so √1351 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1351 has exactly 4 factors.
  • The factors of 1351 are outlined with their factor pair partners in the graphic below.

1351 is the hypotenuse of a Pythagorean triple:
665-1176-1351 which is 7 times (95-168-193)

1350 Logic is at the Heart of This Puzzle

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Today’s Puzzle:

By simply changing two clues of that recently published puzzle that I rejected, I was able to create a love-ly puzzle that can be solved entirely by logic. Can you figure out where to put the numbers from 1 to 12 in each of the four outlined areas that divide the puzzle into four equal sections? If you can, my heart might just skip a beat!

If you need some tips on how to get started on this puzzle, check out this video:

Factors of 1350:

  • 1350 is a composite number.
  • Prime factorization: 1350 = 2 × 3 × 3 × 3 × 5 × 5, which can be written 1350 = 2 × 3³ × 5²
  • The exponents in the prime factorization are 1, 3 and 2. Adding one to each and multiplying we get (1 + 1)(3 + 1)(2 + 1) = 2 × 4 × 3 = 24. Therefore 1350 has exactly 24 factors.
  • Factors of 1350: 1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 27, 30, 45, 50, 54, 75, 90, 135, 150, 225, 270, 450, 675, 1350
  • Factor pairs: 1350 = 1 × 1350, 2 × 675, 3 × 450, 5 × 270, 6 × 225, 9 × 150, 10 × 135, 15 × 90, 18 × 75, 25 × 54, 27 × 50 or 30 × 45
  • Taking the factor pair with the largest square number factor, we get √1350 = (√225)(√6) = 15√6 ≈ 36.74235

Sum-Difference Puzzles:

6 has two factor pairs. One of those pairs adds up to 5, and the other one subtracts to 5. Put the factors in the appropriate boxes in the first puzzle.

1350 has twelve factor pairs. One of the factor pairs adds up to ­75, and a different one subtracts to 75. 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 about the Number 1350:

1350 is the sum of consecutive prime numbers two ways:
It is the sum of the fourteen prime numbers from 67 to 131, and
673 + 677 = 1350

1350 is the hypotenuse of two Pythagorean triples:
810-1080-1350 which is (3-4-5) times 270
378-1296-1350 which is (7-24-25) times 54

1350 is also the 20th nonagonal number because 20(7 · 20 – 5)/2 = 1350

1348 Coloring Paula Krieg’s Polar Rose

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Paula Beardell Krieg recently wrote about using Desmos to create designs that can be colored by hand or by computer programs like Paint. I like using Paint so with her permission I took a design she made and colored it so I could present it here in this post. I chose colors that make me think of spring because, frankly, I’m ready for winter to be over!

Now I’ll write a little bit about the number 1348:

  • 1348 is a composite number.
  • Prime factorization: 1348 = 2 × 2 × 337, which can be written 1348 = 2² × 337
  • 1348 has at least one exponent greater than 1 in its prime factorization so √1348 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1348 = (√4)(√337) = 2√337
  • The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1348 has exactly 6 factors.
  • The factors of 1348 are outlined with their factor pair partners in the graphic below.

1348 is the sum of two squares:
32² + 18² = 1348

1348 is the hypotenuse of a Pythagorean triple:
700-1152-1348 which is 32² – 18², 2(32)(18), 32² + 18²

1348 is also the short leg in a primitive Pythagorean triple:
1348-454275-454277

 

 

1347 and Level 1

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Which ten division facts do you need to know to solve this puzzle? Seriously, you can do this one!

Print the puzzles or type the solution in this excel file: 10 Factors 1347-1356

If you’re not sure how to solve it, I explain how in this youtube video:

Here are some facts about the number 1347:

  • 1347 is a composite number.
  • Prime factorization: 1347 = 3 × 449
  • 1347 has no exponents greater than 1 in its prime factorization, so √1347 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1347 has exactly 4 factors.
  • The factors of 1347 are outlined with their factor pair partners in the graphic below.

1347 isn’t a Lucas number, but as OEIS.org reminds us, 1, 3, 4, 7 happens to be the first four Lucas numbers. Here’s why:
Start with 1, 3, then . . .
1 + 3 = 4
3 + 4 = 7
So what will be the next Lucas number?

1347 is the hypotenuse of a Pythagorean triple:
840-1053-1347 which is 3 times (280-351-449)

(1346÷2)×3 = 2019 Magic or Square?

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I was inspired to make a 3 × 3 Magic Square where every number is different but the numbers in each row, column and diagonal added together equal the same number, 54:

I made it by taking a regular 3 × 3 Magic Square and adding 13 to each of its numbers.

What inspired me to do that? This magical tweet of a palindromic Magic Square for the Year 2019:

Yeah, I know my magic square isn’t quite as impressive. It might be more square than it is magic. It’s also less impressive than this number 2019 spelled out using fifty-one
4 × 4 Magic Squares.

Maybe you will be more impressed by this magic square that has 2019 as its Magic Sum?

I could make that magic sum because 2019 is divisible by 3. Why is 673 in the center? Because 2019÷3 = 673.

673 × 2 = 1346. I’m sharing these magic squares in this post I’ve numbered 1346. In case you haven’t figured it out (1346÷2)×3 = 2019. Happy New Year!

Here’s more about the number 1346:

  • 1346 is a composite number.
  • Prime factorization: 1346 = 2 × 673
  • 1346 has no exponents greater than 1 in its prime factorization, so √1346 cannot be simplified.
  • The exponents in the prime factorization are 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1346 has exactly 4 factors.
  • The factors of 1346 are outlined with their factor pair partners in the graphic below.

1346 is the hypotenuse of a Pythagorean triple:
770-1104-1346 which is 2 times 385-552-673