Let S = \sum_{k=1}^{82} (-1)^k \cdot k \cdot 7^k. What is the remainder when -S is divided… — GMAT

Let $S = \sum_{k=1}^{82} (-1)^k \cdot k \cdot 7^k$ . What is the remainder when $-S$ is divided by $10$ ?

A
1
B
3
C
7
9Answer
E
0

Answer

The remainder when

-S

is divided by

10

9

Grouping the 82 terms into 20 blocks of 4 shows that each block contributes

6 \pmod{10}

, summing to

120 \equiv 0 \pmod{10}

. The 81st and 82nd terms contribute

3

and

8

respectively, giving

S \equiv 11 \equiv 1 \pmod{10}

. Therefore,

-S \equiv -1 \equiv 9 \pmod{10}

Step-by-Step Solution

Analyze the units digit cyclicity of

7^k \pmod{10}

The sequence of units digits for

7^k

repeats every 4 powers:

7^1 \equiv 7

7^2 \equiv 9

7^3 \equiv 3

7^4 \equiv 1 \pmod{10}

Units digits of powers of 7 follow a repeating cycle of length 4.

Evaluate the terms modulo 10 for a general block of 4 consecutive integers

k = 4m+1, 4m+2, 4m+3, 4m+4

The four terms evaluate to:

(-1)^{4m+1}(4m+1)7^{4m+1} \equiv -(4m+1)(7) \equiv 2m+3 \pmod{10}

;

(-1)^{4m+2}(4m+2)7^{4m+2} \equiv (4m+2)(9) \equiv 6m+8 \pmod{10}

;

(-1)^{4m+3}(4m+3)7^{4m+3} \equiv -(4m+3)(3) \equiv 8m+1 \pmod{10}

; and

(-1)^{4m+4}(4m+4)7^{4m+4} \equiv (4m+4)(1) \equiv 4m+4 \pmod{10}

Modular arithmetic simplifies each term according to its position in the 4-term cycle.

Sum the four terms in a single block of 4.

(2m+3) + (6m+8) + (8m+1) + (4m+4) = 20m + 16 \equiv 6 \pmod{10}.

Every complete block of 4 consecutive terms contributes exactly 6 modulo 10, independent of

m

Calculate the total sum

S \pmod{10}

for

k=1

82

82 terms contain 20 full blocks of 4 terms (

k=1

80

) plus 2 remaining terms (

k=81

and

k=82

). The 20 blocks contribute

20 \times 6 = 120 \equiv 0 \pmod{10}

. For

m=20

k=81

gives

2(20)+3 = 43 \equiv 3 \pmod{10}

, and

k=82

gives

6(20)+8 = 128 \equiv 8 \pmod{10}

. Thus

S \equiv 0 + 3 + 8 = 11 \equiv 1 \pmod{10}

Summing the contributions of all complete blocks and the remaining boundary terms gives

S \pmod{10}

Determine the positive remainder of

-S

when divided by 10.

Since

S \equiv 1 \pmod{10}

-S \equiv -1 \equiv 10 - 1 = 9 \pmod{10}

By definition of integer division, the remainder

r

must satisfy

0 \le r < 10

Key Concept

Units Digit Cyclicity and Modular Arithmetic with Negative Integer Remainders

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