What is the remainder when 3^{42} is divided by 5? — GMAT

What is the remainder when $3^{42}$ is divided by $5$ ?

Answer: 4

The remainder when

3^{42}

is divided by 5 is 4.

The remainders of powers of 3 divided by 5 follow a repeating pattern of length 4: (3, 4, 2, 1). To find the remainder of

3^{42} \div 5

, divide the exponent 42 by 4. Since

42 = 4 \times 10 + 2

, the remainder of the exponent is 2. The 2nd term in the repeating sequence is 4, so

3^{42} \pmod 5 = 4

Determine the remainder pattern for consecutive powers of 3 divided by 5.

The remainders for

3^1, 3^2, 3^3, 3^4, 3^5

are

3, 4, 2, 1, 3

, establishing a cycle of length 4.

Powers of integers divided by a positive integer yield repeating sequence patterns (cyclicity).

Divide the target exponent by the cycle length.

Dividing 42 by 4 gives a quotient of 10 and a remainder of 2.

The remainder indicates the specific term within the 4-step cycle.

Evaluate the value corresponding to the 2nd position in the cycle.

The 2nd term in the cycle (3, 4, 2, 1) is 4.

A remainder of 2 in the exponent position corresponds to the same remainder as

3^2

divided by 5.

Remainders and Units Digit Cyclicity

Question