what is the reminder of 5^501/126?

Answer 1

5^3 = -1 mod 126

5^501 = (5^3)^167 or (-1)^167 mod 126
= -1 mod 126

so reminder = 126-1 or 125

Answer 2

We can write 5^501 = 5^3 * 5^498 = 5^3 * (5^6)^83 = 125 * (15625)^83.

Notice that, for any n, 15625n = 15624n + n = (126)(124)n + n. This means that 15625n differs from n by a multiple of 126. Thus, if we divide 15625n by 126, we get the same remainder as if we were to divide n by 126.

Thus, if we divide 125 * (15625)^83 by 126, we get the same remainder as if we divided 125 * (15625)^82 by 126; which is the same remainder as if we divided 125 * (15625)^81; and so on down, until we get the same remainder as if we divided 125 by 126.

Therefore, the remainder when dividing 5^501 by 126 is 125.

(If you know group theory: Work out 5^501 within Z/126Z, the cyclic group of order 126. The argument is easier.)

Answer 3

Using Euler's theorem, a^(Euler-phi(n)) is congruent to 1 mod n. Since 126 = 2 x 3^2 x 7, Euler-phi(126) = (126)(1/2)(2/3)(6/7) = 36. Then 5^36 is congruent to 1 mod 126.

Since 501 = 33 + (13)(36), 5^501 is congruent to 5^33 is congruent to 125^11 is congruent to (-1)^11 is congruent to -1 is congruent to 125 mod 126.

Answer 4

5^3 = -1 mod 126

5^501 = (5^3)^167 or (-1)^167 mod 126
= -1 mod 126

so reminder = 126-1 or 125

Answer 5

125

Answer 6

125 is the remainder. And I'm not even typing out all the digits from my calculator for the quotient. ;)

Answer 7

reminder or remainder? ang labo non a. make yourself understood please.

Answer 8

what math is that?

Answer 9

sorry don't know