Embibe Experts Solutions for Chapter: Number System, Exercise 5: EXERCISE - 2.5

Find the smallest two-digit number $N$ such that the sum of digits of ${10}^N-N$ is divisible by $170.$

Let $N=12345678910111213\dots \dots 4344$ be the $79$ digit number obtained that is formed by writing the integers from $1$ to $44$ in order, one after the other. The remainder when $N$ is divided by $45$ is $r.$ What is $r^2?$

Let $k={2008}^2+2^{2008}.$ What is the square of the units digit of $k^2+2^k?$

Anna randomly picked five integers from the following list $53, 62, 66, 68, 71, 82, 89$ and discover that the average value of the five integers she picked is still an integer. If two of the integers she picked were $62$ and $89,$ find the mean of the remaining three integers.

If the remainder when $\left(1!\times 1\right)+\left(2!\times 2\right)+\left(3!\times 3\right)+\dots +\left(286!\times 286\right)$ is divided by $2^4\times 5^3$ is $r$, then $r-1900$ is

What is the remainder when $3^0+3^1+3^2+\dots +3^{2009}$ is divided by $8?$

How many distinct four-digit numbers are divisible by $3$ and have $23$ as their last two digits?

What is the last two digits of ${2015}^{2016}-2017?$