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 $9$ ?

Cevap: 1

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

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

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