### Divisibility Rules and 924:

Let’s apply some basic divisibility rules to find some of the factors of 924:

- Like every other counting number, 924 is divisible by 1.
- Since 924 is even, it is divisible by 2.
- 9 + 2 + 4 = 15, a number divisible by 3, so 924 is divisible by 3.
- Its last two digits, 24, is divisible by 4, so 924 is divisible by 4.
- Its last digit isn’t 0 or 5, so 924 is NOT divisible by 5.
- 924 is even and divisible by 3, so it is also divisible by 6.
- Since
**92**-2(**4**) = 84, a number divisible by 7, we know that**924**is also divisible by 7. - Since its last two digits are divisible by 8, and the third to the last digit, 9, is odd, 924 is NOT divisible 8.
- 9 + 2 + 4 = 15, a number not divisible by 9, so 924 is NOT divisible by 9.
- The last digit is not 0, so 924 is NOT divisible by 10.
- 9 – 2 + 4 = 11, so 924 is divisible by 11.

Thus, 1, 2, 3, 4, 6, 7, and 11 are all factors of 924.

### Factor Cake for 924:

You can see its prime factors easily on the outside of its festive prime factor cake:

### Factors of 924:

- 924 is a composite number.
- Prime factorization: 924 = 2 x 2 x 3 x 7 x 11, which can be written 924 = 2² x 3 x 7 x 11
- The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 x 2 = 24. Therefore 924 has exactly 24 factors.
- Factors of 924: 1, 2, 3, 4, 6, 7, 11, 12, 14, 21, 22, 28, 33, 42, 44, 66, 77, 84, 132, 154, 231, 308, 462, 924
- Factor pairs: 924 = 1 x 924, 2 x 462, 3 x 308, 4 x 231, 6 x 154, 7 x 132, 11 x 84, 12 x 77, 14 x 66, 21 x 44, 22 x 42, or 28 x 33,
- Taking the factor pair with the largest square number factor, we get √924 = (√4)(√231) = 2√231 ≈ 30.3973683.

### Sum Difference Puzzle:

924 has twelve factor pairs. One of the factor pairs adds up to 65, and a different one subtracts to 65. If you can identify those factor pairs, then you can solve this puzzle!

### More about the Number 924:

You may have seen one of its many possible factor trees contained in the first frame of this factor tree for 852,852 from my previous post:

924 is in the very center of the 12th row of Pascal’s triangle because 12!/(6!6!) = 924.

924 is the sum of consecutive prime numbers: 461 + 463 = 924

I like 924 written in some other bases:

770 BASE 11

336 BASE 17

220 BASE 21

SS BASE 32, S is 28

S0 BASE 33

924 has several sets of consecutive factors. Besides being divisible by the 1st, 2nd, and 3rd triangular numbers (1, 3, and 6), those consecutive factors mean the following:

- 924 is divisible by the 6th triangular number, 21, which is 6(7)/2.
- 924 is divisible by the 11th triangular number, 66, which is 11(12)/2.
- 924 is divisible by the 21st triangular number, 231, which is 21(22)/2.