Many people fly home or away from home for the holidays. Here’s a puzzle to occupy some of your time while you’re in flight.

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

Now I’ll write a little bit about the number 1329:

• 1329 is a composite number.
• Prime factorization: 1329 = 3 × 443
• 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 1329 has exactly 4 factors.
• Factors of 1329: 1, 3, 443, 1329
• Factor pairs: 1329 = 1 × 1329 or 3 × 443
• 1329 has no square factors that allow its square root to be simplified. √1329 ≈ 36.45545

 

1329 is divisible by 3 because it is made with three consecutive numbers (1, 2, 3) and 9, a number divisible by 3.

1329 looks interesting in some other bases:
It’s 929 in BASE 12,
369 in BASE 20
234 in BASE 25

 

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.

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

Here are some facts about the number 1323:

• 1323 is a composite number.
• 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.
• Factors of 1323: 1, 3, 7, 9, 21, 27, 49, 63, 147, 189, 441, 1323
• Factor pairs: 1323 = 1 × 1323, 3 × 441, 7 × 189, 9 × 147, 21 × 63, or 27 × 49
• Taking the factor pair with the largest square number factor, we get √1323 = (√441)(√3) = 21√3 ≈ 36.37307

 

Here is my favorite of the many possible factor trees for 1323:

 

The first Christmas Star led the wise men to find the Baby Jesus.

This Christmas star can lead you to a better knowledge of all the facts in a basic 1 to 10 multiplication table.

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

Now I’ll share a few facts about the number 1322:

• 1322 is a composite number.
• Prime factorization: 1322 = 2 × 661
• 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

 

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.