Let S = 2^{2026} - 3^{2026} + 4^{2026} - 7^{2026}. What is the remainder when S is divided… — GMAT

Let $S = 2^{2026} - 3^{2026} + 4^{2026} - 7^{2026}$ . What is the remainder when $S$ is divided by $10$ ?

A
0
2Answer
C
4
D
6
E
8

Answer

The remainder when

S

is divided by

10

2

Finding the remainder when an expression is divided by 10 is equivalent to finding the units digit of that expression. The units digits of powers of 2, 3, 4, and 7 repeat in periodic cycles of length 4, 4, 2, and 4, respectively. Since

2026 \equiv 2 \pmod 4

, the units digits correspond to the 2nd term of each cycle:

2^2 \rightarrow 4

3^2 \rightarrow 9

4^2 \rightarrow 6

, and

7^2 \rightarrow 9

. Evaluating the expression yields

4 - 9 + 6 - 9 = -8

. In modular arithmetic, a negative remainder

-8 \pmod{10}

is equivalent to

-8 + 10 = 2

. Therefore, the value representing 2 is correct.

Step-by-Step Solution

Determine the remainder of each term divided by 10 by finding the units digit cyclicity.

Powers of 2 cycle with period 4 (2, 4, 8, 6). Since

2026 = 4 \times 506 + 2

2^{2026} \equiv 2^2 \equiv 4 \pmod{10}

Dividing an integer by 10 yields a remainder equal to its units digit.

Evaluate the units digits for the remaining terms

3^{2026}

4^{2026}

, and

7^{2026}

Powers of 3 cycle with period 4 (3, 9, 7, 1);

3^{2026} \equiv 3^2 \equiv 9 \pmod{10}

. Powers of 4 cycle with period 2 (4, 6);

4^{2026} \equiv 4^2 \equiv 6 \pmod{10}

. Powers of 7 cycle with period 4 (7, 9, 3, 1);

7^{2026} \equiv 7^2 \equiv 9 \pmod{10}

Each base follows a repeating pattern of units digits when raised to successive positive integer powers.

Substitute the congruent remainder values back into the expression for

S

S \equiv 4 - 9 + 6 - 9 = -8 \pmod{10}

Modular arithmetic operations preserve addition and subtraction equivalences.

Convert the negative result to a standard non-negative remainder.

-8 + 10 = 2

. Thus, the remainder is

2

By definition, the remainder

r

when an integer is divided by

d

must satisfy

0 \leq r < d

Key Concept

Units Digit Cyclicity and Negative Remainder Rules

Estimated Time:2m 30s

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