In number theory, the units digit of a positive integer raised to successive positive inte… — GMAT

In number theory, the units digit of a positive integer raised to successive positive integer powers follows a repeating cyclic pattern. What is the units digit of $4^{25}$ ?

Answer: 4

Answer

The units digit of

4^{25}

is 4.

The units digit of powers of 4 alternates between 4 (for odd powers) and 6 (for even powers). Because 25 is an odd number,

4^{25}

has a units digit of 4.

Step-by-Step Solution

Identify the units digit pattern for powers of 4

The units digits cycle between 4 (for odd exponents) and 6 (for even exponents), giving a cycle length of 2.

Units digits of positive integer powers follow a periodic pattern determined by the base digit.

Determine the parity of the exponent 25

25 is an odd integer (remainder 1 when divided by 2).

The exponent's remainder modulo 2 determines which position in the 2-element cycle [4, 6] the number falls into.

Select the corresponding units digit from the cycle

Since 25 is odd, the units digit is 4.

Odd powers of 4 always have a units digit of 4.

Key Concept

Units Digit Cyclicity

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