For any positive integer n, let f(n) denote the remainder when 13^n + 17^n is divided by 1… — GMAT

For any positive integer $n$ , let $f(n)$ denote the remainder when $13^n + 17^n$ is divided by $10$ . What is the sum of all possible distinct values of $f(n)$ ?

A
$0$
B
$8$
$10$ Answer
D
$18$
E
$20$

Answer

The sum of all possible distinct values is 10.

Finding the remainder when a number is divided by

10

is equivalent to finding its units digit. The units digit of

13^n

follows the repeating pattern

[3, 9, 7, 1]

, and the units digit of

17^n

follows the repeating pattern

[7, 9, 3, 1]

. Adding these term-by-term yields units digits of

(3+7) \rightarrow 0

(9+9) \rightarrow 8

(7+3) \rightarrow 0

, and

(1+1) \rightarrow 2

. The set of distinct remainders is

\{0, 8, 2\}

, and their sum is

0 + 8 + 2 = 10

Step-by-Step Solution

Relate division by 10 to units digits

The remainder when any positive integer is divided by

10

is equal to its units digit. Therefore,

f(n)

is the units digit of

13^n + 17^n

, which depends only on the units digits of

3^n

and

7^n

Dividing by 10 isolates the ones place of an integer.

Determine the units digit pattern for

3^n

and

7^n

The units digits of

3^n

follow a 4-term repeating cycle:

[3, 9, 7, 1]

. The units digits of

7^n

follow a 4-term repeating cycle:

[7, 9, 3, 1]

Powers of integers exhibit periodic patterns in their units digits.

Evaluate

f(n)

for each term in the 4-step pattern

For

n=1

: units digit of

3 + 7 = 10

0

, so

f(1) = 0

.
For

n=2

: units digit of

9 + 9 = 18

8

, so

f(2) = 8

.
For

n=3

: units digit of

7 + 3 = 10

0

, so

f(3) = 0

.
For

n=4

: units digit of

1 + 1 = 2

2

, so

f(4) = 2

Test one full period of length 4 to find all possible outputs.

Find the sum of all distinct values

The distinct values of

f(n)

are

0

8

, and

2

. Their sum is

0 + 8 + 2 = 10

The question asks for the sum of distinct possible remainders.

Key Concept

Units Digit Cyclicity and Modular Arithmetic

Estimated Time:1m 45s

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