Problem Solving

Here is a problem on the GMAT practice questions. I solved it a different way, and need to know if I stumbled upon the answer by accident or if what I did would hold true if a different version of the question was on the real test.

"If X is a positive integer, and if the units digit of X^2 is 9 and the units digit of (X+1)^2 is 4, what is the units digit of (X+2)^2?"

(a) 1
(b) 3
(c) 5
(d) 6
(e) 14

The answer is (a).

The following is how I got it.

(X+1)^2 = 4
(X+1) (X+1) = 4
X^2 +2X +1 = 4
X^2 +2X -3 = 0
(X+3) (X-1) = 0
X = -3, X = 1

While plugging in both a 1 and a -3 for X into (X+1)2 = 4 will work, only -3^2 = 9 (as per the original statement in the question stem). So I plugged in -3 into (X+2)^2 and got 1. And that's the answer.

Now here is their explanation:

"Only numbers ending in 3 or 7 would yield a units digit of 9 when squared. Thus, if 9 is the units digit of X^2, then either 3 or 7 must be the units digit of X.

If the units digit is 3, then X+1= 3+1 = 4. This makes the units digit of )X+1)^2 the units digit of4^2, which is 6.

If, however, the units digit is 7, then X+1 = 7+1 = 8. This makes the units digit of (X+1)^2 the units digit of 8^2, which is 4, as is needed in this problem. Therefore, the units digit of X must be 7.

Thus, the units digit of X+2 is 9. This makes the units digit of (X+2)^2 the units digit of 9^2, which is 1.

The correct answer is A."

So my question is - if the question had been phrased differently using different numbers, would I still have derived the correct answer?

Yes, you would still get to the right answer, but it takes much longer to get there. Creativity is rewarded on this exam and finding different solutions other than algebraic ones will save you time.

Use the rule of cyclicity here.

As the unit digit of the square of first number is 9. Hence, the last digit of the number can wither be 3,7.
But by knowing number+1 square's unit digit is 4. We can ignore 3 as 3+1 is 4 and sq of any number ending with 4 will be having unit digit 6.

Hance, answer will be 7.
And (7+2) = 9.
Sq of number ending with 9 will be having unit digit as 1.

Hence, answer is A.

jzw wrote:Here is a problem on the GMAT practice questions. I solved it a different way, and need to know if I stumbled upon the answer by accident or if what I did would hold true if a different version of the question was on the real test.

"If X is a positive integer, and if the units digit of X^2 is 9 and the units digit of (X+1)^2 is 4, what is the units digit of (X+2)^2?"

(a) 1
(b) 3
(c) 5
(d) 6
(e) 14

The answer is (a).

The following is how I got it.

(X+1)^2 = 4
(X+1) (X+1) = 4
X^2 +2X +1 = 4
X^2 +2X -3 = 0
(X+3) (X-1) = 0
X = -3, X = 1

While plugging in both a 1 and a -3 for X into (X+1)2 = 4 will work, only -3^2 = 9 (as per the original statement in the question stem). So I plugged in -3 into (X+2)^2 and got 1. And that's the answer.

Now here is their explanation:

"Only numbers ending in 3 or 7 would yield a units digit of 9 when squared. Thus, if 9 is the units digit of X^2, then either 3 or 7 must be the units digit of X.

If the units digit is 3, then X+1= 3+1 = 4. This makes the units digit of )X+1)^2 the units digit of4^2, which is 6.

If, however, the units digit is 7, then X+1 = 7+1 = 8. This makes the units digit of (X+1)^2 the units digit of 8^2, which is 4, as is needed in this problem. Therefore, the units digit of X must be 7.

Thus, the units digit of X+2 is 9. This makes the units digit of (X+2)^2 the units digit of 9^2, which is 1.

The correct answer is A."

So my question is - if the question had been phrased differently using different numbers, would I still have derived the correct answer?

Your approach was right.