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1. The remainder , when (15²³ + 23²³) is divided by 19, is

A. 4

B. 15

C. 0

D. 18

Solution:

NOTE: aⁿ + bⁿ is always divisible by (a + b) when n is odd.

So, (15²³ + 23²³) always divisible by 38.
And 38 is a multiple of 19, So, the number which is divisible by 38, is divisible by 19 too.
(15²³ + 23²³) is divisible by 19.
So,

\frac{15^{23} + 23^{23}}{19}

== Remainder ⇒

\frac{15 + 23}{19}

== Remainder ⇒

\frac{38}{19}

== Remainder ⇒ 0

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