When a positive integer n is divided by 4, the remainder is 3. What is the units digit of … — GMAT

When a positive integer $n$ is divided by $4$ , the remainder is $3$ . What is the units digit of the expression $M = 3^{n+1} + 8^{2n+1} - 7^{n+2}$ ?

The units digit of the expression is 6.

Since

n \equiv 3 \pmod{4}

, we substitute into each exponent:

n+1 \equiv 0 \pmod{4}

2n+1 \equiv 3 \pmod{4}

, and

n+2 \equiv 1 \pmod{4}

. The units digit cyclicity patterns give

3^4 \to 1

8^3 \to 2

, and

7^1 \to 7

. Evaluating the combined units digits yields

1 + 2 - 7 = -4

. Adding 10 to obtain a valid units digit gives

6

Express the integer

n

in terms of modular arithmetic

n \equiv 3 \pmod{4}

, which means

n = 4k + 3

for some non-negative integer

k

Units digits of powers repeat in cycles of 4, so exponent behavior depends on exponents modulo 4.

Determine the units digit of

3^{n+1}

Exponent

n+1 = (4k+3)+1 = 4k+4 \equiv 0 \pmod{4}

. The units digit pattern for 3 is

3, 9, 7, 1

. Since exponent mod 4 is 0 (or 4), the units digit is

1

The cyclicity pattern for powers of 3 has period length 4.

Determine the units digit of

8^{2n+1}

Exponent

2n+1 = 2(4k+3)+1 = 8k+7 \equiv 3 \pmod{4}

. The units digit pattern for 8 is

8, 4, 2, 6

. For exponent mod 4 equal to 3, the units digit is

2

The cyclicity pattern for powers of 8 has period length 4.

Determine the units digit of

7^{n+2}

Exponent

n+2 = (4k+3)+2 = 4k+5 \equiv 1 \pmod{4}

. The units digit pattern for 7 is

7, 9, 3, 1

. For exponent mod 4 equal to 1, the units digit is

7

The cyclicity pattern for powers of 7 has period length 4.

Combine the units digits and resolve negative intermediate values

1 + 2 - 7 = -4 \equiv 6 \pmod{10}

. The units digit is

6

Units digits must be non-negative integers from 0 to 9, so a negative result requires adding 10.

Units Digit Cyclicity and Modular Arithmetic

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