For any positive integer n, what is the units digit of the expression S = 7^{4n+1} + 9^{2n… — GMAT

For any positive integer $n$ , what is the units digit of the expression $S = 7^{4n+1} + 9^{2n} + 3^{4n+3}$ ?

A
0
B
3
5Answer
D
7
E
9

Answer

The units digit of the expression is 5.

Each base has a predictable units digit cyclicity. The pattern for powers of

7

7, 9, 3, 1

(period 4), giving a units digit of

7

for exponent

4n+1

. The pattern for powers of

9

9, 1

(period 2), giving a units digit of

1

for even exponent

2n

. The pattern for powers of

3

3, 9, 7, 1

(period 4), giving a units digit of

7

for exponent

4n+3

. Adding these units digits gives

7 + 1 + 7 = 15

, whose units digit is

5

Step-by-Step Solution

Determine the units digit of

7^{4n+1}

The units digit of powers of

7

follows a repeating pattern of length 4:

7, 9, 3, 1

. Since the exponent

4n+1

has a remainder of

1

when divided by

4

, the units digit is

7

Units digit cyclicity of base

7

repeats every 4 powers.

Determine the units digit of

9^{2n}

The units digit of powers of

9

follows a repeating pattern of length 2:

9, 1

. For any positive integer

n

, the exponent

2n

is even, so the units digit is

1

Even powers of

9

always end in

1

Determine the units digit of

3^{4n+3}

The units digit of powers of

3

follows a repeating pattern of length 4:

3, 9, 7, 1

. Since the exponent

4n+3

has a remainder of

3

when divided by

4

, the units digit is

7

Units digit cyclicity of base

3

repeats every 4 powers.

Sum the units digits and take the final units digit

7 + 1 + 7 = 15

, which has a units digit of

5

The units digit of a sum of numbers equals the units digit of the sum of their individual units digits.

Key Concept

Units digit cyclicity of integer powers

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic