Consider the number obtained by multiplying 48 and 53. If this product is divided by 50, w… — State PSC Exam

Consider the number obtained by multiplying $48$ and $53$ . If this product is divided by $50$ , what is the resulting positive remainder?

Answer: 44

Applying the properties of modular arithmetic,

48

is congruent to

-2 \pmod{50}

and

53

is congruent to

3 \pmod{50}

. Multiplying these gives a remainder of

-6 \pmod{50}

. To find the standard positive remainder, we add the divisor

50

-6

, resulting in

44

. Alternatively, direct calculation gives

48 \times 53 = 2544

, and

2544

divided by

50

yields a quotient of

50

with a remainder of

44

Find the remainder of 48 divided by 50.

-2

Using a negative remainder simplifies the multiplication step.

Find the remainder of 53 divided by 50.

Individual remainders are needed to apply the product rule of modular arithmetic.

Multiply the individual remainders.

-2 \times 3 = -6

The remainder of a product is congruent to the product of the individual remainders.

Convert the negative remainder to a positive remainder.

50 - 6 = 44

A standard remainder must be a non-negative integer strictly less than the divisor.

Remainder Theorem and Negative Modulus

Question