What is the remainder when the mathematical expression 14 \times 47^{55} - 63 is divided b… — State PSC Exam

What is the remainder when the mathematical expression $14 \times 47^{55} - 63$ is divided by $48$ ?

The correct remainder is 19.

The correct answer is found by applying modular arithmetic properties. First, simplify the base:

47 \equiv -1 \pmod{48}

. Raising this to an odd power gives

47^{55} \equiv (-1)^{55} = -1 \pmod{48}

. Substituting this back into the expression yields

14 \times (-1) - 63

. We can also simplify

63 \pmod{48}

15

. The expression then becomes

-14 - 15 = -29 \pmod{48}

. Because standard remainders cannot be negative, we add the divisor (48) to find the true positive remainder:

-29 + 48 = 19

Simplify the base of the exponent modulo 48.

47 \equiv -1 \pmod{48}

Using negative remainders makes calculating large powers manageable.

Evaluate the exponentiated term modulo 48.

(-1)^{55} = -1

An odd power of -1 remains -1.

Simplify the constant term modulo 48.

63 \equiv 15 \pmod{48}

To keep all terms within the modulo base before finalizing the expression.

Substitute the simplified values back into the expression.

14 \times (-1) - 15 = -14 - 15 = -29

To find the overall remainder of the expression, which currently results in a negative value.

Convert the negative remainder to a valid positive remainder.

-29 + 48 = 19

Remainders must be strictly positive integers less than the divisor.

Divisibility Rules and Remainder Theorem

Soru