Embibe Experts Solutions for Chapter: Number System, Exercise 1: EXERCISE - 2.1

Suppose that $m$ and $n$ are positive integers such that $75m=n^3$. What is the minimum possible value of $m+n$?

Let $n$ be the largest integer that is the product of exactly $3$ distinct prime numbers $d, e$ and $10d+e$, where $d$ and $e$ are single digits. What is the sum of the digits of $n$ ?

The digits $1, 2, 3, 4, 5, 6, 7$ and $9$ are used to form four two-digit prime numbers, with each digit used exactly once, then the sum of these four prime numbers is given by $10A$. Find the value of $A$

The positive integers $A, B, A-B$ and $A+B$ are all prime numbers. The sum of these four prime numbers is

Find the number of counterexamples to the statement:

"If $N$ is an odd positive integer, the sum of whose digits is $4$ and none of whose digits is $0,$ then $N$ is prime."

Three positive integers, each greater than $1,$ have a product of $27000$ and are pairwise relatively prime. What is their mean to the nearest whole number?

A positive integer $n$ is nice if there is a positive integer $m$ with exactly four positive divisors (including $1$ and $m$) such that the sum of the four divisors is equal to $n$. How many numbers in the set $\left\{2010, 2011, 2012, \dots ..., 2019\right\}$ are nice?

Let $a$ and $b$ be relatively prime integers with $a>b>0$ and $\frac{a^3-b^3}{{\left(a-b\right)}^3}=\frac{73}{3}$. What is $a-b$?