Nishit K Sinha Solutions for Chapter: Numbers and Their Properties, Exercise 2: Benchmarking Test 1

A number has exactly $32$ factors; from these factors, $4$ are not composite. The product of these $4$ factors is $30$. How many such numbers are possible?

The first $126$natural numbers are given side by side in ascending order to form a large number: $123,456\dots 125,126$. What will be the remainder when this large number is divided by $5625$?

${1000}^7-17$ is written in decimal notation. What is the sum of its digits?

$\mathrm{N}$ is an integer. For how many distinct values of $\mathrm{N}$, ${\mathrm{N}}^4-20{\mathrm{N}}^2+4$ is a prime number?

Of $128$ boxes of oranges, each box contains at least $120$ and at most $144$ oranges. The number of boxes containing the same number of oranges is at least:

What is the value of $\sum _{\mathrm{N}=1}^{99}\frac{1}{\sqrt{\mathrm{N}}+\sqrt{\mathrm{N}+1}}$?

$\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}$ and $\mathrm{e}$ are prime numbers, not necessarily consecutive. If $\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d}+\mathrm{e}=350$ and $\mathrm{a}<\mathrm{b}<\mathrm{c}<\mathrm{d}<\mathrm{e}$, how many values of ${\mathrm{a}}^5$ is/are possible?

What is the remainder when $2^{104}$ is divided by $101$?