GMAT Math : Modulo arithmetic

Modulo arithmetic is a very powerful tool for divisibility and remainder type questions.

First, an example:

What is the remainder when t² + 2t + 3 is divided by 8?

1. t when divided by 8 yields a remainder of 1

2. t – 3 when divided by 8 yields a remainder of 1

Fundamentally, the basic notation in modulo arithmetic is as follows

x = a mod (n)  which means when number x is divided by number n the remainder is a.

3 = 1 mod (2) is an example of the above notation.

The really cool thing about modulo arithmetic is the ability to add, subtract and even raise to powers these notations, provided they have the same base ‘n’. Specifically :

if x = a mod (n)

and z = b mod (n)

The above two equations can be added to get (x + z) = (a + b) mod (n)

They can be subtracted to get (x – z) = (a – b) mod (n)

They can be multiplied by a constant term xk = ak mod (n) (where k is a constant)

They can be raised to powers xt = at mod (n)

But what does this all mean and how is this useful? If you see a question like what is the remainder when 10099 is divided by 3 – modulo arithmetic can solve it easily.

We know that 100 when divided by 3 gives a remainder of 1 or in modulo arithmetic terms

100 = 1 mod (3). Using the rule for powers and raising both sides of the equation by 99.

10199 = 199 mod (3)

10199 = 1 mod (3)

This tells us that the remainder when 10099 is divided by 3 is 1. Pretty nifty huh?

Getting back to the illustrative question.

Statement (1) can be represented as t = 1 mod (8) so

t² = 1² mod (8) = 1 mod (8)

2t = 2 x 1 mod (8) = 2 mod (8)

3 = 3 mod (8)

Adding the above three terms we arrive at the question statement

t² + 2t + 3 = 6 mod (8).

In other words, t² + 2t + 3 when divided by 8 gives a remainder of 6. so statement (1) is sufficient.

Statement (2) tells us that t² = 1 mod (8) or specifically t = 1 mod (8) or t = -1 mod (8). Notice this statement has not been able to tell us the remainder when t is divided by 8 – it can be 1 or -1 (-1 means a remainder of 7). And we need the remainder when t is divided by 8 to find the remainder of the second term (2t) in the original question statement. So information is insufficient. We go with Option (A). For a more detailed exposition of modulo arithmetic and its utility to solve divisibility data sufficiency questions, please do check out our Hackbook