What is the remainder when 4^{15} is divided by 5? — State PSC Exam

What is the remainder when $4^{15}$ is divided by $5$ ?

The remainder when

4^{15}

is divided by

5

4

Since

4 \equiv -1 \pmod 5

, raising to the 15th power gives

(-1)^{15} = -1 \pmod 5

. Adding the divisor

5

converts the negative remainder

-1

into the valid positive remainder

4

Express the base number in modular form relative to the divisor.

4 = 5 - 1

, which means

4 \equiv -1 \pmod 5

Expressing the base as

-1

modulo

5

simplifies exponent calculation.

Raise the modular congruence to the power of 15.

4^{15} \equiv (-1)^{15} \pmod 5 = -1 \pmod 5

An odd power of

-1

evaluates to

-1

Convert the negative remainder into a positive remainder.

-1 + 5 = 4

Standard remainders in integer division must be non-negative integers less than the divisor.

Remainder Theorem and Negative Remainder Conversion

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