1410 and Level 1

Start the school year off right with a quick review of the multiplication table. You can actually construct an entire 10 × 10 table with only the nine clues in this puzzle. Figure out where the numbers 1 to 10 go in both the first column and the top row and amaze yourself with how much you remember!

  • 1410 is a composite number.
  • Prime factorization: 1410 = 2 × 3 × 5 × 47
  • 1410 has no exponents greater than 1 in its prime factorization, so √1410 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1410 has exactly 16 factors.
  • The factors of 1410 are outlined with their factor pair partners in the graphic below.

1410 is the hypotenuse of a Pythagorean triple:
846-1128-1410 which is (3-4-5) times 282.

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1391 and Level 1

Many of the clues in this puzzle have double digits. If you know why they do, then you can find all the factors and solve this puzzle!

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Here’s some information about the number 1391:

  • 1391 is a composite number.
  • Prime factorization: 1391 = 13 × 107
  • 1391 has no exponents greater than 1 in its prime factorization, so √1391 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 1391 has exactly 4 factors.
  • The factors of 1391 are outlined with their factor pair partners in the graphic below.

1391 is the hypotenuse of a Pythagorean triple:
535-1284-1391 which is (5-12-13) times 107

1373 and Level 1

This puzzle is really just a multiplication table but the factors are missing and the products are not in order. You can figure out where the factors go, and then the clues will all make sense.


Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Now I’ll share some facts about the puzzle number, 1373:

  • 1373 is a prime number.
  • Prime factorization: 1373 is prime.
  • 1373 has no exponents greater than 1 in its prime factorization, so √1373 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1373 has exactly 2 factors.
  • The factors of 1373 are outlined with their factor pair partners in the graphic below.

How do we know that 1373 is a prime number? If 1373 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1373. Since 1373 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 31, or 37, we know that 1373 is a prime number.

1373 is the sum of two squares:
37² + 2² = 1373

1373 is the hypotenuse of a Pythagorean triple:
148-1365-1373 calculated from 2(37)(2), 37² – 2², 37² + 2²

Here’s another way we know that 1373 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 37² + 2² = 1373 with 37 and 2 having no common prime factors, 1373 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1373. Since 1373 is not divisible by 5, 13, 17,  29, or 37, we know that 1373 is a prime number.

1357 and Level 1

Do you know the multiplication and division facts necessary to solve this puzzle? If you don’t, this puzzle will give you much-needed practice, so give it a try!

Print the puzzles or type the solution in this excel file: 12 Factors 1357-1365

Here is some information about the puzzle number, 1357:

  • 1357 is a composite number.
  • Prime factorization: 1357 = 23 × 59
  • 1357 has no exponents greater than 1 in its prime factorization, so √1357 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 1357 has exactly 4 factors.
  • The factors of 1357 are outlined with their factor pair partners in the graphic below.

1, 3, 5, 7 are the first four odd numbers in order.

1342 Level One Puzzles Only

My son requested I made some pages using only level 1 puzzles for my granddaughter to do. You can see and use the first page above, but I’ve also compiled ALL the level 1 puzzles from 2018 and put them in one excel file: Level 1’s from 2018

Those puzzles will give beginners lots of practice and help them learn the multiplication table.

Now I’ll tell you a little bit about the number 1342:

  • 1342 is a composite number.
  • Prime factorization: 1342 = 2 × 11 × 61
  • 1342 has no exponents greater than 1 in its prime factorization, so √1342 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 1342 has exactly 8 factors.
  • The factors of 1342 are outlined with their factor pairs in the graphic below.

1342 is the hypotenuse of a Pythagorean triple:
242-1320-1342 which is 22 times (11-60-61)

And (11-60-61) times 122 is a triple where 1342 is a leg:
1342-7320-7442. Pretty cool!

The last prime number was 1327 which was 15 numbers ago. The next prime number won’t be until 1361 which is 19 numbers away. Wow! That means that 1342 is 15 numbers away from the CLOSEST prime number, a new record. Thank you Stetson.edu for alerting me to that fact.

1333 and Level 1

How well do you know the multiplication table? If you don’t know it very well, this puzzle can be a fun, no-pressure way to get more familiar with it. Give it a try!

Print the puzzles or type the solution in this excel file: 12 factors 1333-1341

Now I’ll share some facts about the puzzle number, 1333:

  • 1333 is a composite number.
  • Prime factorization: 1333 = 31 × 43
  • 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 1333 has exactly 4 factors.
  • Factors of 1333: 1, 31, 43, 1333
  • Factor pairs: 1333 = 1 × 1333 or 31 × 43
  • 1333 has no square factors that allow its square root to be simplified. √1333 ≈ 36.51027

1333 is the difference of two squares two different ways:
667²-666² = 1333
37² – 6² = 1333

1321 Christmas Gift Puzzle

 

This puzzle looks like a gift because it is! You can fill out an entire multiplication table just from the 16 clues in this puzzle.

Print the puzzles or type the solution in this excel file:10-factors-1321-1332

Here are a few facts about the number 1321:

  • 1321 is a prime number.
  • Prime factorization: 1321 is prime.
  • The exponent of prime number 1321 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1321 has exactly 2 factors.
  • Factors of 1321: 1, 1321
  • Factor pairs: 1321 = 1 × 1321
  • 1321 has no square factors that allow its square root to be simplified. √1321 ≈ 36.34556

How do we know that 1321 is a prime number? If 1321 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1321 ≈ 36.3. Since 1321 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1321 is a prime number.

 

1321 is the sum of nine consecutive prime numbers:
127 + 131 + 137 + 139 + 149 + 151 + 157 + 163 + 167 = 1321

1321 is the sum of two squares:
36² +  5² = 1321

1321 is the hypotenuse of a primitive Pythagorean triple:
360-1271-1321 calculated from 2(36)(5), 36² –  5², 36² +  5²

Here’s another way we know that 1321 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 36² + 5² = 1321 with 36 and 5 having no common prime factors, 1321 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1321 ≈ 36.3. Since 1321 is not divisible by 5, 13, 17, or 29, we know that 1321 is a prime number.

1311 Little Square Candies

Here’s a puzzle made with some sweet squares. The nine clues in it are all you need to find the factors and complete the entire “mixed-up” multiplication table. 

Print the puzzles or type the solution in this excel file: 12 factors 1311-1319

Now I’ll write some facts about the number 1311:

  • 1311 is a composite number.
  • Prime factorization: 1311 = 3 × 19 × 23
  • 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 1311 has exactly 8 factors.
  • Factors of 1311: 1, 3, 19, 23, 57, 69, 437, 1311
  • Factor pairs: 1311 = 1 × 1311, 3 × 437, 19 × 69, or 23 × 57
  • 1311 has no square factors that allow its square root to be simplified. √1311≈ 36.20773

1311 is the sum of five consecutive prime numbers:
251 + 257 + 263 + 269 + 271 = 1311

As shown in their factor trees below, 1308, 1309, 1310, and 1311 each have three distinct prime numbers in their prime factorizations. They are the smallest set of four consecutive numbers with the same number of prime factors. 1309, 1310, and 1311 are also the smallest three consecutive numbers that have exactly the same number of factors and factor pairs. Thank you Stetson.edu for alerting me to those facts.

Finally, although the list of primes is slightly out of order, here’s a fun fact related to 1311 that I saw on Twitter:

 

 

1302 and Level 1

If you can tell time on an analog clock, then you can solve this puzzle. Go ahead and see how little time it will take you!

Print the puzzles or type the solution in this excel file: 10-factors-1302-1310

Here are a few facts about the number 1302:

  • 1302 is a composite number.
  • Prime factorization: 1302 = 2 × 3 × 7 × 31
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1302 has exactly 16 factors.
  • Factors of 1302: 1, 2, 3, 6, 7, 14, 21, 31, 42, 62, 93, 186, 217, 434, 651, 1302
  • Factor pairs: 1302 = 1 × 1302, 2 × 651, 3 × 434, 6 × 217, 7 × 186, 14 × 93, 21 × 62, or 31 × 42
  • 1302 has no square factors that allow its square root to be simplified. √1302 ≈ 36.08324

1289 and Level 1

You might think this is a very easy puzzle, but for some people, it will be challenging, and will hopefully help them learn some multiplication facts better.

Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

Since this is puzzle number 1289, I’ll share some facts about that number:

  • 1289 is a prime number.
  • Prime factorization: 1289 is prime.
  • The exponent of prime number 1289 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1289 has exactly 2 factors.
  • Factors of 1289: 1, 1289
  • Factor pairs: 1289 = 1 × 1289
  • 1289 has no square factors that allow its square root to be simplified. √1289 ≈ 35.90265

How do we know that 1289 is a prime number? If 1289 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1289 ≈ 35.9. Since 1289 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1289 is a prime number.

1289 is the sum of two squares:
35² + 8² = 1289

1289 is the hypotenuse of a Pythagorean triple:
560-1161-1289 calculated from 2(35)(8), 35² – 8², 35² + 8²

Here’s another way we know that 1289 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 35² + 8² = 1289 with 35 and 8 having no common prime factors, 1289 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1289 ≈ 35.9. Since 1289 is not divisible by 5, 13, 17, or 29, we know that 1289 is a prime number.