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 $10$ ?

Cevap: 5

Cevap

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.

Adım Adım Çözüm

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

Anahtar Kavram

Units Digit Cyclicity and Modular Arithmetic

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