Evaluate the remainder when the numeric expression 47^{101} + 111^{101} - 25 is divided by… — State PSC Exam

Evaluate the remainder when the numeric expression $47^{101} + 111^{101} - 25$ is divided by $16$ .

Cevap: 5

By applying the properties of modular arithmetic, we determine that

47 \equiv -1 \pmod{16}

and

111 \equiv -1 \pmod{16}

. Raising these to the 101st power yields

-1

and

-1

. Subtracting 25 gives a total of

-27

. To find the positive remainder, we add the next highest multiple of 16 (which is 32) to -27, yielding a final valid positive remainder of 5.

Find the remainder of the bases when divided by 16.

47 \equiv -1 \pmod{16}

and

111 \equiv -1 \pmod{16}

Using negative remainders (since

47 = 16 \times 3 - 1

and

111 = 16 \times 7 - 1

) significantly simplifies the calculation of large powers.

Substitute the simplified bases into the original expression.

(47)^{101} + (111)^{101} - 25 \equiv (-1)^{101} + (-1)^{101} - 25 \pmod{16}

The properties of modular arithmetic allow replacing bases with their congruent values modulo the divisor.

Evaluate the exponents and the constant term.

(-1) + (-1) - 25 = -27

An odd power of a negative number remains negative, so

(-1)^{101} = -1

Find the equivalent positive remainder for -27 modulo 16.

-27 \equiv 5 \pmod{16}

Adding multiples of 16 (in this case,

16 \times 2 = 32

) to

-27

yields a positive remainder in the valid range of

0

15

. Calculating

-27 + 32 = 5

gives the final remainder.

Modular arithmetic with large powers using negative remainders

Instead of converting the bases to -1, one could convert them to positive 15. The expression becomes

15^{101} + 15^{101} - 25

. However, evaluating this requires recognizing that

15 \equiv -1 \pmod{16}

to easily compute the large exponents, which ultimately merges back into the primary solution path.

Tahmini Süre:1m 30s

Soru