2.5 SAT Math - Divisibility Rules

Good shortcuts to know for the SAT math section:

A number is divisible by	If
2*	it's even
3*	the sum of the digits is a multiple of 3
4	the last two digits are divisible by 4
5*	it ends in 5 or 0
6*	the sum of the digits is a multiple of 3, and it's even
8	the sum of the last three digits are divisible by 8
9	the sum of the digits is a multiple of 9
10*	it ends in 0

*These are the more useful.  Unless you know the others already or are really, really good at memorizing, use your calculator for the others and all other integers not listed here.

Which brings us to:

-Shortcut Strategy: Cross-canceling, common fractions/decimal equivalents, and divisibility rules are all excellent shortcuts that it will serve you well to know and use.  However, if you don't see the shortcut immediately, do the work on your calculator! Students get stubborn when it comes to the SATs.  They get a gut instinct that tells them there should be a quicker and easier way to do the question, and they spend an exorbitant amount of time trying to remember what that way is.  You want to know what you're doing when that happens? You're having fun.  On the SATs.  You are.  Your brain is enjoying trying to figure out a little puzzle like you're at the beach with your grandma doing Sudoku.  Do not have fun on the SATs.  Answer the question as quickly as you can in the first accurate way that comes to mind, and move on.  If you want to play with fun little shortcuts later, feel free to go home and do some math after the test.

Remainders and Patterns

When you first learned how to divide, if you remember, you did something like this:
                                           

And then you'd take that remainder 2 and write your answer as "5 R 2" Right?

As you got older, you started taking that remainder 2 and putting it on top of the divisor, so your answer became “

5	2 --------- 3

.”  Then you started adding zeros to the end of the dividend to get a decimal, so you'd end up with "5.66" or "5.667" as an answer.  When you're dealing with remainders on the SAT exam, you need to go back to early-level math, the "R 2" kind.  Your calculator, however, doesn't know this.  Your calculator is going to keep giving you decimals.  Luckily there's a way around this.

Let's say the question asks for the remainder when 13,802 is divided by 13.  You could do the long division to get that whole number remainder, but that will take a while. 

Better, type "13802/13" into your calculator.  What do you get? 1061.69231.  Your calculator created a decimal out of the remainder, but your answer needs to be that whole number "R" that’s leftover.  So what do you do?

1) Subtract the integer to the left of the decimal.  That is, get rid of the main "answer" part.  So in this case, you'll subtract 1061.  You're now left with just the decimal portion of the answer.

2) Multiply by what you originally divided by.  This is because your calculator took that whole number remainder and stuck it in top of your original divisor, so we need to "undo" that work.  So in this case, we multiply .69231 by 13 to undo that operation.  What do you get? 9.  That's your remainder.  If you'd done the long division it would have looked like this:
                                        

-Hint: If the question asks for a remainder the answer is the whole number that is left over, NOT the fraction, NOT the decimal, and certainly NOT the full answer.  So using the same example question above, what is the correct answer? 

What is the remainder when 13802 is divided by 13?
(A) .69231


(B)	9 --------- 13


(C) 9
(D) 1061
(E) 1061.69231

The answer is (C).

A trickier type of remainder question actually gives you the remainder and asks you to find one of the original numbers used in the division problem.  For example:

When a positive integer x is divided by 6, the remainder is 3.  What is the remainder when 3x is divided by six?

The easiest way to do these types is to pick a multiple of the divisor and add the specified remainder to that number.  This will be your "original" that you'll substitute in wherever you see the variable.  So in the example above, pick a multiple of 6.  Let's say 12.  Add the specified remainder, which here is 3.  You get 15.  Now, wherever you see x, use "15" instead.  So 3x is actually "3×15", which is 45.  So what's the remainder when 45 is divided by six?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

The answer is (C) 3. 

Back to SAT Math Study Guide

Next 2.6 SAT Math - Patterns