n is a positive integer greater than 2. If y = 9^0 + 9^1 + 9^2 + + 9

Question

n is a positive integer greater than 2. If y = 9^0 + 9^1 + 9^2 + … + 9^n, what is the remainder when y is divided by 5?

(1) n is divisible by 3.
(2) n is odd

GMATinsight · Accepted Answer

CONCEPT  : The ramainder when any number is divided by 5 is the remainder when the unit digit of the Number is divided by 5 therefore such questions require the calculation of the Unit digit of the number (y in this case) only

Question Redefined: y = 9^0 + 9^1 + 9^2 + … + 9^n

Statement 1: n is divisible by 3

@n=3, y = 9^0 + 9^1 + 9^2 + 9^3 = Unit digits are   = Unit digit of the numbers 0 i.e. Remainder 0 when y is divided by 5
@n=6, y = 9^0 + 9^1 + 9^2 +---+ 9^6 = Unit digits are   = Unit digit of the numbers 1 i.e. Remainder 1 when y is divided by 5

Hence,   NOT SUFFICIENT

Statement 2: n is odd

@n=3, y = 9^0 + 9^1 + 9^2 + 9^3 = Unit digits are   = Unit digit of the numbers 0 i.e. Remainder 0 when y is divided by 5
@n=5, y = 9^0 + 9^1 + 9^2 + 9^3 + 9^4 + 9^5 = Unit digits are   = Unit digit of the numbers 0 i.e. Remainder 0 when y is divided by 5

with the above working, we can observe that y will always have even number of terms for every odd value of n as the minimum power starts from 0 in each terms.

Also  every pair of two consecutive terms in y gives us 0 Unit digit  i.e. sum of all pairs of terms in y becomes 0 and the remainder when y is divided by 5 will also become 0

Hence,   SUFFICIENT

Answer: option  B

Bunuel · Answer

MANHATTAN GMAT OFFICIAL SOLUTION: Remember the units digit pattern for 9^x, where x is an integer. The units digit of 9^x is 9 if x is odd, but the units digit is 1 if x is even: a repeating pattern of . Now, consider the sums of the powers of 9 up to 9^n: 2015-06-15_1629.png The alternating 1's and 9's in the units digits pair to a sum of 10, or a units digit of 0. Thus, the units digit of the sum displays another two-term repeating pattern. The units digit of y is 0 if n is odd, but 1 if n is even. The remainder when y is divided by 5 depends only on the units digit and will be either 0 or 1 as well. So the rephrased question is “Is n odd or even?” (1) INSUFFICIENT: If n is a multiple of 3, it may be either odd or even. (2) SUFFICIENT: If n is odd, the units digit of y is 0, and the remainder is 0 when y is divided by 5. The correct answer is B.

broall · Answer

$9 \equiv -1 \pmod{5} \implies 9^k \equiv (-1)^k \pmod{5}$

(1) $n=3k$ so

$\begin{split}
y &\equiv (-1)^0 &+(-1)^1 &+(-1)^2 &+...&+(-1)^{3k} &\pmod{5}\
y &\equiv \quad 1 &+ (-1) &+\quad 1 &+...&+(-1)^k &\pmod{5}\
\end{split}$

If $k$ is even, for example $k=2$, then $y \equiv 1 \pmod{5}$
If $k$ is odd, for example $k=1$, then $y \equiv 0 \pmod{5}$

Insufficient.

(2) $n=2k+1$ so
$\begin{split}
y &\equiv (-1)^0 &+(-1)^1 &+(-1)^2 &+...&+(-1)^{2k}&+(-1)^{2k+1} &\pmod{5}\
y &\equiv \quad 1 &+ (-1) &+\quad 1 &+...&+\quad 1 &+(-1) &\pmod{5}\
y &\equiv \quad 0 &\pmod{5}
\end{split}$

Sufficient

Hence the answer is B

Other solution  
$y \equiv (-1)^0 +(-1)^1 +(-1)^2 +...+(-1)^{n} = \frac{(-1)^{n+1}-1}{(-1)-1}=\frac{-(-1)^{n}-1}{-2}=\frac{(-1)^n+1}{2}\pmod{5}$

(1) If $n=3k \implies y \equiv \frac{(-1)^{3k}+1}{2} \pmod{5}$
If $k=1 \implies y \equiv \frac{(-1)^3+1}{2}=0 \pmod{5}$
If $k=2 \implies y \equiv \frac{(-1)^6+1}{2}=1 \pmod{5}$

Hence insufficient.

(2) If $n=2k+1 \implies y \equiv \frac{(-1)^{2k+1}+1}{2}=\frac{(-1)+1}{2}=0 \pmod{5}$
Hence sufficient.

ShashankDave · Answer

Hi  Bunuel ...
I used negative remainders for this one...
9 when divided by 5 gives out a remainder of -1.
Lets see this given series by replacing 9 with -1...

(-1)^0 + (-1)^1 + (-1)^2 +(-1)^3..... +(-1)^n

So if n is odd, we will have the sum equal to 0.
And if n is even, we will have the sum equal to 1.

Now let's check the statements..
(1) n could be either odd or even. Not sufficient.
(2) N is odd. Sufficient

Answer (B)

coolkl · Answer

"Y = 1+ 9+9^2... y = 5k + r. R = 0 if we get 0 or 5 as units digit

S-1) n = 3, 6,9..
If we see the sequence if n = 3
1+ 9 + 81+ 729 is div by 5
n = 6 => consider only units digit 1 + 9 + 1 + 9 + 1+ 9 +1 not div by 5.
NS

S-2) Sufficient if n is always odd n = 3,5 as in S-1
If we see the sequence if n = 3
1+ 9 + 81+ 729 is div by 5
"
Answer is B

bumpbot · Answer

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n is a positive integer greater than 2. If y = 9^0 + 9^1 + 9^2 + + 9

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n is a positive integer greater than 2. If y = 9^0 + 9^1 + 9^2 + + 9

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