If a and b are positive integers, what is the remainder...

[AAPL]

If a and b are positive integers, what is the remainder when 9^(2a+1+b) is divided by 10?

(1) a = 3
(2) b is odd.

The OA is B.

I need help with this DS question. Please, can any expert explain it for me? Thanks.

[Jay@ManhattanReview]

If a and b are positive integers, what is the remainder when 9^(2a+1+b) is divided by 10?

(1) a = 3
(2) b is odd.

The OA is B.

I need help with this DS question. Please, can any expert explain it for me? Thanks.

We have 9^(2a+1+b) = (3^2)^(2a+1+b) = 3^[2.(2a+1+b)] = 3^(4a+2+2b)

Thus, we are asked to determine the remainder when 3^(4a+2+2b) is divided by 10.

The remainder will be governed by the unit digit of 3^(4a+2+2b).

Let's understand the power cycle of 3.

1. 3^1 = 3; the unit digit = 3;
2. 3^2 = 9; the unit digit = 9;
3. 3^3 = 27; the unit digit = 7;
4. 3^4 = 81; the unit digit = 1;

5. 3^5 = 243; the unit digit = the unit digit of 3^(4 + 1) = 3;
6. 3^6 = 243; the unit digit = the unit digit of 3^(4 + 2) = 9;
7. 3^7 = 2187; the unit digit = the unit digit of 3^(4 + 3) = 7;
8. 3^8 = 6561; the unit digit = the unit digit of 3^(4 + 4) = 1

We see that the unit digit of the power of 3 follows a cycle of 4: 3, 9, 7, and 1.

Thus,

9. The unit digit of 3^(4n + 1) = the unit digit of 3^1 = 3; ignore 4n;
10. The unit digit of 3^(4n + 2) = the unit digit of 3^2 = 9; ignore 4n;
11. The unit digit of 3^(4n + 3) = the unit digit of 3^3 = 7; ignore 4n;
12. The unit digit of 3^(4n + 4) = the unit digit of 3^4 = 1; ignore 4n.

Let's switch to the question.

Question rephrased: What is the unit digit of 3^(4a+2+2b)?

We have 3^(4a+2+2b). The unit digit of 3^(4a+2+2b) = the unit digit of 3^(2+2b); ignore 4a.

(1) a = 3

Since a has no role to play, and we do not have the value of b, the statement is not sufficient,

(2) b is odd.

We have 3^(2+2b) = 3^2(1+b) = 3^2(1 + ODD) = 3^2(EVEN) = 3^(a multiple of 4)

The unit digit of 3^(a multiple of 4) = the unit digit of 3^4 = 1. Sufficient.

The correct answer: B

Hope this helps!

-Jay
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[Brent@GMATPrepNow]

If a and b are positive integers, what is the remainder when 9^(2a+1+b) is divided by 10?

(1) a = 3
(2) b is odd.

Target question: What is the remainder when 9^(2a+1+b) is divided by 10?
This is a great candidate for rephrasing the target question.

First recognize that this is a clever way of asking, "What is the units digit of 9^(2a+1+b)?"
Notice that 153 divided by 10 equals 15 with remainder 3
Likewise, 3218 divided by 10 equals 321 with remainder 8
And 97 divided by 10 equals 9 with remainder 7
So, we can write....
REPHRASED target question: What is the units digit of 9^(2a+1+b)?

IMPORTANT: We can RE-rephrase this target question in a way that makes it super easy to analyze the statements.
To see how, let's examine some powers of 9
9^1 = 9
9^2 = 81
9^3 = 729
9^4 = 6561
.
.
.
Notice that, when the exponent is ODD, the units digit is 9
When the exponent is EVEN, the units digit is 1
So, all we need to do is determine whether or not the exponent, (2a+1+b), is ODD or EVEN
To make things easier, we should recognize that 2a is EVEN for all integer values of a.
This means 2a+1 is ODD for all integer values of a.
So, if b is ODD, then 2a+1+b = ODD + ODD = EVEN, which means the units digit of 9^(2a+1+b) is 1
And, if b is EVEN, then 2a+1+b = ODD + EVEN = ODD, which means the units digit of 9^(2a+1+b) is 9
So, to answer the target question, all we need to know is whether b is odd or even
So,.......
RE-REPHRASED target question: Is n even or odd?

Aside: Here's a video with tips on rephrasing the target question: https://www.gmatprepnow.com/module/gmat- ... cy?id=1100

Statement 1: a = 3
This is not enough information to determine whether n is even or odd
Since we cannot answer the RE-REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: b is odd
Perfect!!
Since we can answer the RE-REPHRASED target question with certainty, statement 2 is SUFFICIENT

Answer: B

Cheers,
Brent