For any positive integer n, let S_n denote the units digit of the sum 2^n + 3^n + 4^n + 7^… — GMAT

For any positive integer $n$ , let $S_n$ denote the units digit of the sum $2^n + 3^n + 4^n + 7^n$ . What is the remainder when the sum $T = \sum_{n=1}^{102} S_n$ is divided by $9$ ?

Answer: 1

Answer

The remainder when the sum T is divided by 9 is 1.

Evaluating the units digits of each exponent term reveals that

S_n

follows a repeating 4-term cycle

(6, 8, 2, 4)

with a sum of 20 per cycle. For 102 terms, there are 25 full cycles plus the first two terms (

S_1=6

and

S_2=8

). The total sum is

T = 25 \times 20 + 6 + 8 = 514

. Dividing 514 by 9 yields

514 = 9 \times 57 + 1

, so the remainder is 1.

Step-by-Step Solution

Find the cyclicity of the units digit of each term

2^n, 3^n, 4^n,

and

7^n

Units digits repeat in patterns of length 4: for

2^n

(2, 4, 8, 6), for

3^n

(3, 9, 7, 1), for

4^n

(4, 6, 4, 6), and for

7^n

(7, 9, 3, 1).

Units digits of positive integer powers cycle with periodicities that divide 4.

Compute

S_n

for

n = 1, 2, 3, 4

and find the sum of one 4-term period.

S_1 = \text{units}(2+3+4+7=16) = 6

S_2 = \text{units}(4+9+6+9=28) = 8

S_3 = \text{units}(8+7+4+3=22) = 2

S_4 = \text{units}(6+1+6+1=14) = 4

. Sum of one period =

6+8+2+4 = 20

The sum of the units digits of individual terms determines the units digit of the total expression.

Calculate the total sum

T = \sum_{n=1}^{102} S_n

Since

102 = 25 \times 4 + 2

, the sequence consists of 25 complete cycles of 4 terms plus the first 2 terms (

S_1=6

and

S_2=8

). Thus,

T = 25 \times 20 + 6 + 8 = 500 + 14 = 514

Dividing the total number of terms by the period length gives the number of full cycles and remaining initial terms.

Compute the remainder when

T = 514

is divided by 9.

The sum of the digits of 514 is

5 + 1 + 4 = 10

, and

10 \equiv 1 \pmod 9

. Alternatively,

514 = 9 \times 57 + 1

. Thus, the remainder is 1.

A positive integer and the sum of its digits leave the same remainder when divided by 9.

Key Concept

Units digit cyclicity of exponential terms and modular arithmetic on sequence sums

Estimated Time:2m 0s

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