When the positive integer n is divided by 12, the remainder is 7. What is the units digit … — GMAT

When the positive integer $n$ is divided by $12$ , the remainder is $7$ . What is the units digit of $9^n + 4^{n+1} - 7^{n+2}$ ?

A
2
B
4
C
6
8Cevap
E
0

Cevap

The units digit of the expression is 8.

The correct answer is 8 because evaluating each component using unit digit cyclicity gives

9^n ≡ 9 \pmod{10}

(since

n

is odd),

4^{n+1} ≡ 6 \pmod{10}

(since

n+1

is even), and

7^{n+2} ≡ 7 \pmod{10}

(since

n+2 ≡ 1 \pmod 4

). Combining these yields

(9 + 6 - 7) = 8

Adım Adım Çözüm

Express

n

using division algorithm and determine its properties.

Since

n = 12k + 7

for some non-negative integer

k

n

is odd,

n+1

is even, and

n+2 = 12k + 9

Establishing the form of

n

determines the exponents for cyclicity calculations.

Find the units digit of

9^n

Units digit of

9^n

is 9.

Powers of 9 alternate units digits:

9^1 = 9, 9^2 = 1, 9^3 = 9...

Any odd power of 9 ends in 9. Since

n = 12k+7

is odd,

9^n

ends in 9.

Find the units digit of

4^{n+1}

Units digit of

4^{n+1}

is 6.

Powers of 4 alternate units digits:

4^1 = 4, 4^2 = 6, 4^3 = 4...

Any even power of 4 ends in 6. Since

n

is odd,

n+1

is even, so

4^{n+1}

ends in 6.

Find the units digit of

7^{n+2}

Units digit of

7^{n+2}

is 7.

Powers of 7 follow a 4-step cyclicity pattern: 7, 9, 3, 1. The exponent

n+2 = 12k + 9 = 4(3k+2) + 1 ≡ 1 \pmod 4

. Thus,

7^{n+2}

has the same units digit as

7^1

, which is 7.

Combine the units digits.

Units digit =

9 + 6 - 7 = 8

Adding and subtracting the respective units digits gives

15 - 7 = 8

Anahtar Kavram

Units Digit Cyclicity and Modular Arithmetic

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