# Math: Divisibility Rules

**Q1**. Which of the following is NOT divisible by 3?

(A) 231

(B) 246

(C) 285

(D) 326

(E) 411

One way to do this problem is to divide each answer choice by 3 and see which one leaves a remainder. A far better approach, however, is to apply the following rule of divisibility for 3. If the sum of the digits of any number is a multiple of 3, then that digit is divisible by 3.

Let’s try to apply this rule below:

111: 1 + 1 + 1 = 3. Three is a multiple of three so 111 is divisible by three.

3,456: 3 + 4 + 5 + 6 = 18. Eighteen is a multiple of 3 so 3,456 is divisible by three.

2,789: 2 + 7 + 8 + 9 = 26. Twenty-six is NOT a multiple of three. Therefore, 2,789 is not divisible by three.

Now let’s return to the original question.

(A) 231: 2 + 3 + 1 = 6 (Multiple)

(B) 246: 2 + 4 + 6 = 12 (Multiple)

(C) 285: 2 + 8 + 5 = 15 (Multiple)

(D) 326: 3 + 2 + 6 = 11 (Not Multiple) Answer

(E) 411: 4 + 1 + 1 = 6 (Multiple)

**Answer: B**.

The next important rule to know is the rule of divisibility by 4: If the last two digits of a number are a multiple of 4, then the number is divisible by 4.

724: the last two digits are 24. 24 is a multiple of 4. (Divisible)

470: 70 is not a multiple of 4 (\dfrac{70}{4} = 17.5). (Not Divisible)

40,004: 04, or 4 is a multiple of 4 (Divisible)

Now let’s try the following problem in which you have to combine everything you learned today:

**Q2**. Which of the following integers is divisible by 12?

(A) 1,442

(B) 1,653

(C) 1,728

(D) 2,048

(E) 2,884

What, you gasp. Divisibility of 12. You never taught us that! Well, for a number to be divisible by 12, it has to be divisible by both 4 and 3, because 4 \times 3 = 12.

(A) 1,442: 42 is not a multiple of 4.

(B) 1,653: 53 is not a multiple of 4 (though the sum of the digits is divisible by 3).

(C) 1,728: 28 is a multiple of 4; the sum of the digits, 18, is a multiple of 3.

(D) 2,048: sum of digits 14. Not a multiple of 3.

(E) 2,884 : sum of digits is 22. Not a multiple of 3.

**Answer: C**.