We have 9^(2a+1+b) = (3^2)^(2a+1+b) = 3^[2.(2a+1+b)] = 3^(4a+2+2b)AAPL wrote: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.
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
_________________
Manhattan Review GMAT Prep
Locations: New York | Singapore | Doha | Lausanne | and many more...
Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.
