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Finding Remainder of Numbers of the Form a^b

If 599 is div...

Question

If $5^{99}$ is divided by $13$ , then the remainder is

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Solution

Given
$5^{99} = 5 (5)^{98} = 5 (25)^{49} = 5 (26 - 1)^{49} = 5 [^{49} C_{0} (26)^{49} -^{49} C_{1} (26)^{48} +^{49} C_{2} (26)^{47} + \dots - 1] = 5 [26 k - 1], k \in Z$
Now,
$\frac{5^{99}}{13} = \frac{5 [26 k - 1]}{13} = 10 k - \frac{5}{13} = 10 k - 1 + \frac{8}{13}$

Hence, the remainder is $8$ .

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Finding Remainder of Numbers of the Form a^b

Standard XII Mathematics

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