For any positive integer k, what is the units digit of the expression 2^{4k+2} + 3^{4k+3} … — GMAT

For any positive integer $k$ , what is the units digit of the expression $2^{4k+2} + 3^{4k+3} + 7^{4k+1}$ ?

To find the units digit of

2^{4k+2} + 3^{4k+3} + 7^{4k+1}

, analyze the units digit cyclicity of each base. Powers of 2 have a units digit cycle of [2, 4, 8, 6]. Since

4k+2

leaves a remainder of 2 when divided by 4,

2^{4k+2}

ends in 4. Powers of 3 have a units digit cycle of [3, 9, 7, 1]. Since

4k+3

leaves a remainder of 3 when divided by 4,

3^{4k+3}

ends in 7. Powers of 7 have a units digit cycle of [7, 9, 3, 1]. Since

4k+1

leaves a remainder of 1 when divided by 4,

7^{4k+1}

ends in 7. Adding these units digits gives

4 + 7 + 7 = 18

, so the final units digit is 8.

Determine the units digit cyclicity pattern for base 2

The units digits of powers of 2 repeat in a 4-step cycle: 2, 4, 8, 6. For

2^{4k+2}

, the exponent leaves a remainder of 2 when divided by 4, so its units digit is 4.

Units digits of powers follow periodic cycles modulo 10.

Determine the units digit cyclicity pattern for base 3

The units digits of powers of 3 repeat in a 4-step cycle: 3, 9, 7, 1. For

3^{4k+3}

, the exponent leaves a remainder of 3 when divided by 4, so its units digit is 7.

The exponent

4k+3

corresponds to the 3rd position in the 4-step cycle.

Determine the units digit cyclicity pattern for base 7

The units digits of powers of 7 repeat in a 4-step cycle: 7, 9, 3, 1. For

7^{4k+1}

, the exponent leaves a remainder of 1 when divided by 4, so its units digit is 7.

The exponent

4k+1

corresponds to the 1st position in the 4-step cycle.

Sum the units digits and take the units digit of the result

Sum =

4 + 7 + 7 = 18

, which has a units digit of 8.

The units digit of a sum of integers depends only on the sum of their individual units digits.

Units Digit Cyclicity of Exponents

Soru