When the mathematical expression 44^{63} - 17^2 is divided by 45, what is the final positi… — State PSC Exam

When the mathematical expression $44^{63} - 17^2$ is divided by $45$ , what is the final positive remainder?

Answer: 25

Applying modular arithmetic rules, the base

44

is congruent to

-1 \pmod{45}

. Raising

-1

to an odd power (

63

) keeps the value as

-1

. For the second term,

17^2

equals

289

, which leaves a remainder of

19

when divided by

45

. Subtracting the second remainder from the first gives

-1 - 19 = -20

. Because a standard remainder must be positive, adding the divisor (

45

) to

-20

yields the final correct answer of

25

Apply modular arithmetic to the first term,

44^{63}

, relative to the divisor

45

44 \equiv -1 \pmod{45}

, so

44^{63} \equiv (-1)^{63} = -1 \pmod{45}

Using negative remainders for bases close to the divisor significantly simplifies large power calculations.

Evaluate the second term,

17^2

, and find its remainder when divided by

45

17^2 = 289

. Dividing

289

45

yields a quotient of

6

(

270

) with a remainder of

19

The constant term must be reduced modulo 45 to properly combine it with the first term.

Combine the simplified terms according to the original expression structure.

-1 - 19 = -20 \pmod{45}

The remainder of a difference is equivalent to the difference of the individual remainders.

Convert the resulting negative remainder into an equivalent positive remainder.

-20 + 45 = 25

Standard remainders must be non-negative. Adding the divisor to a negative remainder provides the mathematically correct positive value.

Modular Arithmetic and Negative Remainders

Question