1326 is the 51st triangular number because 26 × 51 = 1326 and 2(26) is 1 more than 51.
Since 51 is an odd number, 1326 is also a hexagonal number. It is the 26th hexagonal number because 26 is half of 52 which is 1 more than 51.
Triangular numbers aren’t hard to visualize, but hexagonal numbers might be. Here are 1326 tiny dots showing that 1326 is a hexagonal number:
Here are some more facts about the number 1326:
1326 is a composite number.
Prime factorization: 1326 = 2 × 3 × 13 × 17
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 1326 has exactly 16 factors.
1326 has no square factors that allow its square root to be simplified. √1326 ≈ 36.41428
Here are some factor trees for 1326 that won’t topple over.
1326 is also the long side of four different Pythagorean triple triangles:
840-1026-1326 which is 6 times (140-171-221)
126-1320-1326 which is 6 times (21-220-221)
510-1224-1326 which is (5-12-13) times 102
624-1170-1326 which is (8-15-17) times 78
Prime factorization: 1325 = 5 × 5 × 53, which can be written 1325 = 5² × 53
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 1325 has exactly 6 factors.
Taking the factor pair with the largest square number factor, we get √1325 = (√25)(√53) = 5√53 ≈ 36.40055
1325 is the sum of two squares in three different ways: 29² + 22² = 1325 34² + 13² = 1325 35² + 10² = 1325
1325 is the hypotenuse of SEVEN Pythagorean triples: 115-1320-1325 which is 5 times (23-264-265) 357-1276-1325 calculated from 29² – 22², 2(29)(22), 29² + 22² 371-1272-1325 which is (7-24-25) times 53 480-1235-1325 which is 5 times (96-247-265) 700-1125-1325 calculated from 2(35)(10), 35² – 10², 35² + 10² 795-1060-1325 which is (3-4-5) times 265 884-987-1325 calculated from 2(34)(13), 34² – 13², 34² + 13²
The Gingerbread man can be tricky so be careful while solving this puzzle. He has fooled and run away from many different people and animals. The mystery is can YOU outfox this one?
The factoring information for the number 1324 is far less tricky than you might imagine:
1324 is a composite number.
Prime factorization: 1324 = 2 × 2 × 331, which can be written 1324 = 2² × 331
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 1324 has exactly 6 factors.
If you start at the top of this Christmas tree puzzle and work your way down, you can find all the factors that make this puzzle work like a multiplication table.
Prime factorization: 1323 = 3 × 3 × 3 × 7 × 7, which can be written 1323 = 3³ × 7²
The exponents in the prime factorization are 3 and 2. Adding one to each and multiplying we get (3 + 1)(2 + 1) = 4 × 3 = 12. Therefore 1323 has exactly 12 factors.
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 1322 has exactly 4 factors.
Factors of 1322: 1, 2, 661, 1322
Factor pairs: 1322 = 1 × 1322 or 2 × 661
1322 has no square factors that allow its square root to be simplified. √1322 ≈ 36.35932
1322 is the sum of six consecutive prime numbers: 199 + 211 + 223 + 227 + 229 + 233 = 1322
1322 is the sum of two squares: 31² + 19² = 1322
1322 is the hypotenuse of a Pythagorean triple: 600-1178-1322 calculated from 31² – 19², 2(31)(19), 31² + 19²
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.
Regardless of its size, an evergreen tree is a mighty symbol at Christmastime. Today’s factoring puzzle features a couple of relatively small Christmas trees, but don’t think for even one minute that these little trees make for an easy puzzle. It’s a level 6 puzzle so there are several places that the clues might trick you. Use logic throughout the entire process, and you should be able to solve it!
That factoring puzzle has nothing to do with the factors of 1320. In case you are looking for factor trees for the number 1320, here are a few of the MANY possible ones:
Factors of 1320:
1320 is a composite number.
Prime factorization: 1320 = 2 × 2 × 2 × 3 × 5 × 11, which can be written 1320 = 2³ × 3 × 5 × 11.
The exponents in the prime factorization are 3, 1, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) )(1 + 1) = 4 × 2 × 2 × 2 = 32. Therefore 1320 has exactly 32 factors.
Taking the factor pair with the largest square number factor, we get √1320 = (√4)(√330) = 2√330 ≈ 36.331804
Sum-Difference Puzzles:
330 has eight factor pairs. One of those pairs adds up to 61, and another one subtracts to 61. Put the factors in the appropriate boxes in the first puzzle.
1320 has sixteen factor pairs. One of the factor pairs adds up to 122, and a different one subtracts to 122. 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?
1320 is the third number that is at the top of more than one Sum-Difference Puzzle. This next one is a primitive.
Again, 1320 has sixteen factor pairs. One of the factor pairs adds up to 73, and a different one subtracts to 73. If you can identify those factor pairs, then you can solve this puzzle!
If you would like a little help finding those factor pairs that make sum-difference, the chart below lists all of 1320’s factor pairs with their sums and their differences.
More about the Number 1320:
1320 is the sum of consecutive primes FOUR different ways: It is the sum of the eighteen primes from 37 to 109, 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 = 1320, 149 + 151 + 157 + 163 + 167 + 173 + 179 + 181 = 1320, and 659 + 661 = 1320.
1320 is the hypotenuse of a Pythagorean triple: 792-1056-1320 which is (3-4-5) times 264.
There are nineteen clues in this challenge puzzle, including two 50’s, two 20’s, two 8’s, and two 12’s. Some of those duplicates might make it more difficult for you to find the one and only solution to the puzzle. I’m very curious about how you do with it!
The exponent of prime number 1319 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1319 has exactly 2 factors.
Factors of 1319: 1, 1319
Factor pairs: 1319 = 1 × 1319
1319 has no square factors that allow its square root to be simplified. √1319 ≈ 36.31804
How do we know that 1319 is a prime number? If 1319 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1319 ≈ 36.3. Since 1319 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1319 is a prime number.
Candy canes are rarely alone. They almost always have a twin close-by. Nevertheless, this mystery-level candy cane puzzle only looks similar to the previous one. You will have to consider completely different factors to solve it.
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 1318 has exactly 4 factors.
Factors of 1318: 1, 2, 659, 1318
Factor pairs: 1318 = 1 × 1318 or 2 × 659
1318 has no square factors that allow its square root to be simplified. √1318 ≈ 36.30427
Candy canes are a sweet tradition every Christmas. This mystery level puzzle won’t be easy, but it can be a sweet way to remember all the multiplication and division facts needed to solve it.
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 1317 has exactly 4 factors.
Factors of 1317: 1, 3, 439, 1317
Factor pairs: 1317 = 1 × 1317 or 3 × 439
1317 has no square factors that allow its square root to be simplified. √1317 ≈ 36.29049
1317 is a palindrome in several bases: It’s 10100100101 in BASE 2, 919 in BASE 12, 525 in BASE 16, and 1J1 in BASE 28,
