Question

The remainder when ${7^{103}}$ is divided by 25 is
A.7
B.24
C.18
D.1

Answer 1

Hint: This is a multi-concept problem of divisibility and binomial expansion. The binomial expansion of ${\left( {a + bx} \right)^n}$ is given by-
${\left( {a + bx} \right)^n} = {}_{}^nC_0^{}{a^n} + {}_{}^nC_1^{}{a^{n - 1}}b + {}_{}^nC_2^{}{a^{n - 2}}{b^2} + ... + {}_{}^nC_n^{}{a^n}$

Complete step by step answer:
To find the remainder the number when divided by 25, we can write in in the form of
25k + r, where the value of r will be the remainder.

${7^{103}} = 7\left( {{7^{102}}} \right) = 7{\left( {{7^2}} \right)^{51}}$
$ = 7{\left( {49} \right)^{51}}$
$ = 7{\left( {50 - 1} \right)^{51}}$
Using the formula for binomial expansion-
$ = 7\left( {{}_{}^{51}C_0^{}{{50}^{51}} + {}_{}^{51}C_1^{}{{50}^{50}}{{\left( { - 1} \right)}^1} + {}_{}^{51}C_2^{}{{50}^{49}}{{\left( { - 1} \right)}^2} + ... + {}_{}^{51}C_{51}^{}{{50}^0}{{\left( { - 1} \right)}^{51}}} \right)$

Taking 50 common in all the terms excluding the last term,
$ = 7\left[ {\left( {50} \right)\left( {{}_{}^{51}C_0^{}{{50}^{50}} + {}_{}^{51}C_1^{}{{50}^{49}}{{\left( { - 1} \right)}^1} + {}_{}^{51}C_2^{}{{50}^{48}}{{\left( { - 1} \right)}^2} + ... + {}_{}^{51}C_0^{}{{50}^0}{{\left( { - 1} \right)}^{50}}} \right) - 1} \right]$
$ = 7\left[ {\left( {50} \right)\left( {{}_{}^{51}C_0^{}{{50}^{50}} + {}_{}^{51}C_1^{}{{50}^{49}}{{\left( { - 1} \right)}^1} + {}_{}^{51}C_2^{}{{50}^{48}}{{\left( { - 1} \right)}^2} + ... + {}_{}^{51}C_0^{}{{50}^0}{{\left( { - 1} \right)}^{50}}} \right)} \right] - 7$
$ = 7\left( {25} \right)\left( 2 \right)\left( {{}_{}^{51}C_0^{}{{50}^{50}} + {}_{}^{51}C_1^{}{{50}^{49}}{{\left( { - 1} \right)}^1} + {}_{}^{51}C_2^{}{{50}^{48}}{{\left( { - 1} \right)}^2} + ... + {}_{}^{51}C_0^{}{{50}^0}{{\left( { - 1} \right)}^{50}}} \right) - 7$
$ = 25k - 7,\;where\;k = 7\left( 2 \right)\left( {{}_{}^{51}C_0^{}{{50}^{50}} + {}_{}^{51}C_1^{}{{50}^{49}}{{\left( { - 1} \right)}^1} + {}_{}^{51}C_2^{}{{50}^{48}}{{\left( { - 1} \right)}^2} + ... + {}_{}^{51}C_0^{}{{50}^0}{{\left( { - 1} \right)}^{50}}} \right)$

We need to bring this in the form 25k + r, so-
= 25k - (25 - 18)
= 25(k - 1) + 18

Hence, the remainder is 18.
The correct option is C. 18

Note: Most students write the answer as 7, because they get 25k - 7. But this is incorrect. We have to bring the expression in the form 25 + r, where r is a positive integer. But at 25k - 7, the remainder becomes negative, which is not possible.