The integers r, s and t all have the same remainder when divided by 5. What is the value of t? GMAT Explanation + Additional Practice!

The integers r, s and t all have the same remainder when divided by 5. What is the value of t?

(1) r + s = t

(2) 20 <= t <= 24

Here’s a challenging remainder question from the GMAT official practice tests. We’re going to start with a very intuitive down to earth approach with a helpful remainders shortcut and then do an algebraic solution.

Define the question: What is the value of t?

There’s not much else to define here but it’s important to note that we are only dealing with integers. 

 

Organize the given information:

Integers  r, s, and t  all have the same remainder when divided by 5. That gives a starting point to organize things.Let’s break this down a bit.. To keep things simple let’s pick some numbers to flesh out what those different remainders could be.

 

5 (reminder 0 when divided by 5)

6 (reminder 1 when divided by 5)

7 (reminder 2 when divided by 5)

8 (reminder 3 when divided by 5)

9 (reminder 4 when divided by 5)

10 (back to 0)

 

In GMAT tutoring sessions we recommend using remainder columns like this:

Remainder 0	Remainder 1	Remainder 2	Remainder 3	Remainder 4
5	6	7	8	9
10	11	12	13	14
15	16	17	18	19
20	21	22	23	24

 

This provides a map for the remainders and will help you spot patterns. Notice that to get to the next number in the same remainder column you add the number you are dividing by. 

There are a bunch of tricks that can be applied with this chart that we won’t go into here because they’re not needed to solve the question.

So when divided by 5 you end up with 5 reminder possibilities, 0-4. That pattern is the same for every number. For 6 the remainder options are 0-5. For 7, 0-6. And so on.

 

R, S, and T are all either:

Remainder 0, 1, 2, 3, or 4.

 

That doesn’t limit them to one value though. There are an infinite number of numbers that are remainder 0 when divided by 5.

So R could be 5, S 10, and T 15. Or they could all be 5. The same goes for all of the other remainder options.

Now that we have some context let’s look at the statements.

 

(1) r + s = t

Does r + s = t and our remainder information limit the value t?

I’d refer to the reminder columns chart.

Remainder 0	Remainder 1	Remainder 2	Remainder 3	Remainder 4
5	6	7	8	9
10	11	12	13	14
15	16	17	18	19
20	21	22	23	24

 

You should see that there are a bunch of options that satisfy the equation and provide a different value for t (5 + 5 = 10, 10 + 10 = 20). 

T could be an infinite number of numbers. Statement 1 is insufficient. 

 

(2) 20 <= t <= 24

Now we have some information severely limiting the options for t.:

Let’s list them out: 20, 21, 22, 23, 24

Does the fact that r, s, and t have the same remainder when divided by 5 help us narrow down this list? Nope. t could be any of those numbers. 

 

Remainder 0	Remainder 1	Remainder 2	Remainder 3	Remainder 4
5	6	7	8	9
10	11	12	13	14
15	16	17	18	19
20	21	22	23	24

If you’re not sure about that take a look at the chart. If t is 20 then r and s could be any number in that first column.

If t is 21 then r and s could be any number in the second column.

Statement 2 is insufficient.

 

Let’s put both statements together:

T is either 20, 21, 22, 23, or 24.

And

R + S = T

And 

They all have the same remainder when divided by 5

 

Once again:

Remainder 0	Remainder 1	Remainder 2	Remainder 3	Remainder 4
5	6	7	8	9
10	11	12	13	14
15	16	17	18	19
20	21	22	23	24

 

T has to be one of the numbers in the last row (statement 2).

R and S must be numbers in the same column as t (given) that sum to t (statement 1).

Let’s just go column by column and see in which columns we can make that work.

 

Column 1

T = 20

R = 10

S = 10

That works. 

 

Looking at the rest of the columns there are no possible r and s values that will sum to to corresponding t. Let’s look at column 2 just to make this clear.

T = 21

Our options for r and s  are: 6, 11, and 16. No combination of those equals 21 so t cannot equal 21. 

Only column 1 works so T must be 20. We need both statements.

Answer: C.

 

Algebraic Solution: The integers r, s and t all have the same remainder when divided by 5. What is the value of t?

Let’s call our remainder ‘n’. Since this is a remainder when dividing an integer by 5, we know n must be either 0, 1, 2, 3, or 4.

Now, let’s look at our integers r, s, and t. These each have the same remainder n when divided by 5. This equivalently means they are all ‘n’ greater than some multiple of 5. So let’s find a way to write this in math-terms. 

 

r = 5(some integer) + n

s = 5(some integer) + n

t = 5(some integer) + n

 

Let’s make this more explicit by assigning names to these arbitrary integers — x, y, and z seem as good a choice as any.

 

r = 5x + n

s = 5y + n

t = 5z + n

 

Now that we have our information organized, let’s bring in our additional statements.

 

(1) r + s = t

Let’s substitute in our x, y, and z equations to see what additional information that gives us.

5x + 5y + 2n = 5z + n

We can use algebra to solve for the remainder n in terms of x, y, and z.

n = 5z – 5x – 5y

n = 5(z – x – y)

 

While we don’t know the values of x, y, or z, this equation tells us something very important — n is a multiple of 5! How do we know? Well, x, y, and z are all integers, meaning the quantity ‘z – x – y’ is also an integer. Therefore, n is just 5 times some integer, or in other words, a multiple of 5.

Let’s think back to the possible values for n: 0, 1, 2, 3, or 4. Of these, the only value that’s a multiple of 5 is 0. Therefore, statement 1 let’s us deduce that our remainder is 0.

However, this leaves us with t = 5z, and since we still don’t know the value of z, we don’t have enough information to solve for t. So we know statement 1 alone is not sufficient.

 

(2) 20 <= t <= 24

We can also think back to our equation for t in terms of z:

t = 5z + n

What else do we know here? Well, we know that the remainder must be between 0 and 4 (Without statement 1, we can’t say the remainder is 0 anymore). Therefore, we know z must be 4! This would give us t = 20 + n, and with a remainder between 0 and 4, this means 20 <= t <= 24, which is what we know from statement 2.

However, with just statement 2 alone, we cannot figure out what this remainder should be. Therefore, statement 2 alone is insufficient.

Let’s look at both equations together:

Using our equation for t as:

t = 5z + n

 

From statement (1), we learned that the value of n must be 0. From statement (2), we learned that the value of z must be 4. Therefore, plugging in these values, we get t = 20. Therefore, both statements together are sufficient, and neither statement alone was sufficient.

Answer: C.