A and B each have some coins. The number of coins A has is the square

Question

A and B each have some coins. The number of coins A has is the square of number of coins that B has. The total number of coins they have between them is a multiple of 100. What is the smallest total number of coins they could have?

A. 200
B. 300
C. 400
D. 500
E. 600

A. 200
B. 300
C. 400
D. 500
E. 600

LeoN88 · Answer

I did manually, since we know the options wrote x and added with x^2 to see if it's 200, 300, 400 or  500

22 + 484

x2+x=600
x(x+1)=600= 24*25

firas92 · Answer

Total = x^2 + x

Only when total=600, we get integer roots for the above equation

x^2+x=600 
 x(x+1)=600

x=24 
And  x^2=576

Answer is  (E)

Hit Kudos if this helped!

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IanStewart · Answer

The number of coins is x^2 + x = x(x+1). So we're multiplying two consecutive integers, x and x+1, and getting a multiple of 100. At most one of two consecutive integers can be divisible by 5, but we need (x)(x+1) to be a multiple of 100 = (2^2)(5^2), so a simple multiple of 5 won't do. We need one of our two numbers to be a multiple of 5^2 = 25. So the smallest possible value for one of our two consecutive integers is 25, and the smallest possible product is (24)(25) = 600.

BraydinWalters · Answer

The first time around I started writing all numbers with their squares and wasted time. Here's how I solved the second time:

You're adding the number and it's square  and it must be a multiple of 100.

9 * 9 = 81 + 9 = 90 X
10 * 10 = 100 + 10 = 110 X

It must be greater than 100. But guessing which numbers to multiply next could be time consuming if you don't have tables memorized.

I now realized that  the units digits must add up to 0 for it to be a multiple of 100 . Check units digits:

1 * 1 will have a units digit of 1. Adding 1 will give you a units digit of 2 which will not be divisible by 100.

Example: 21 * 21 = 441 and 441 + 21 = 462

Based on this, we see  the units digit of the original number must end in a 4, 5, or 9

...4 * ...4 = ...6 + ...4 = ...0
...5 * ...5 = ...5 + ...5 = ...0
...9 * ...9 + ...1 + ...9 = ...0

With practice you'll get the feel for where to go next but if not, start with teens:

14 * 14 = 196 + 14 = 210  not a multiple of 100 
15 * 15 = 225 + 15 = 240  not a multiple of 100  
19 * 19 = 361 + 19 = 380  not a multiple of 100

Now move onto the next tens digit (2) and repeat

24 * 24 = 576 + 24 = 600

E is your answer

A and B each have some coins. The number of coins A has is the square

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GMAT Problem Solving (PS) Questions

A and B each have some coins. The number of coins A has is the square

Prep Toolkit

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