What is the remainder when the expression 7^{45} \cdot 3^{22} + 8^{15} is divided by 10? — GMAT

What is the remainder when the expression $7^{45} \cdot 3^{22} + 8^{15}$ is divided by $10$ ?

Answer: 5

Answer

The remainder is 5.

Dividing any number by 10 leaves a remainder equal to the units digit of that number. By analyzing the units digit pattern (cyclicity of period 4) for powers of 7, 3, and 8:

7^{45} \equiv 7^1 = 7 \pmod{10}

3^{22} \equiv 3^2 = 9 \pmod{10}

, and

8^{15} \equiv 8^3 = 2 \pmod{10}

. The expression simplifies to

(7 \times 9) + 2 = 63 + 2 = 65

, which has a units digit of 5. Therefore, the remainder when divided by 10 is 5.

Step-by-Step Solution

Relate remainder modulo 10 to units digit cyclicity.

Finding the remainder when an expression is divided by 10 is equivalent to finding its units digit.

Any positive integer

N

can be expressed as

10k + r

, where

r

is the units digit and the remainder when

N

is divided by 10.

Determine the units digit of

7^{45}

7^{45} \equiv 7 \pmod{10}

Powers of 7 repeat their units digits in a cycle of length 4 (7, 9, 3, 1). Dividing the exponent 45 by 4 gives a remainder of 1, so

7^{45}

has the same units digit as

7^1 = 7

Determine the units digit of

3^{22}

3^{22} \equiv 9 \pmod{10}

Powers of 3 repeat their units digits in a cycle of length 4 (3, 9, 7, 1). Dividing the exponent 22 by 4 gives a remainder of 2, so

3^{22}

has the same units digit as

3^2 = 9

Determine the units digit of the product

7^{45} \cdot 3^{22}

7^{45} \cdot 3^{22} \equiv 3 \pmod{10}

The product of the units digits is

7 \times 9 = 63

, which has a units digit of 3.

Determine the units digit of

8^{15}

8^{15} \equiv 2 \pmod{10}

Powers of 8 repeat their units digits in a cycle of length 4 (8, 4, 2, 6). Dividing the exponent 15 by 4 gives a remainder of 3, so

8^{15}

has the same units digit as

8^3 = 512

, which ends in 2.

Combine the results to find the final remainder modulo 10.

Adding the units digit of the first term (3) and the second term (2) gives

3 + 2 = 5

Key Concept

Units Digit Cyclicity and Modular Arithmetic

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