Embibe Experts Solutions for Chapter: Number Theory, Exercise 1: IOQM - PRMO and RMO 2019 (25-08-2019)

If $a, b, c$ are real numbers and $(a+b-5{)}^2+(b+2c+3{)}^2+(c+3a-10{)}^2=0$, find the integer nearest to $a^3+b^3+c^3$.

Find the number of positive integers $n$ such that the highest power of $7$ dividing $n!$ is $8.$

Find the largest $2$-digit number $N$ which is divisible by $4,$ such that all integral power of $N$ end with $N.$

The prime numbers $a, b$ and $c$ are such that $a+b^2=4c^2.$ Determine the sum of all possible values of $a+b+c.$

A five-digit number $n=\overline{abcde}$ is such that when divided respectively by $2, 3, 4, 5, 6$ the remainders are $a, b, c, d, e.$ What is the remainder when $n$ is divided by $100?$

Let $a, b, c$ be three distinct positive integers such that the sum of any two of them is a perfect square and having minimal sum $a+b+c.$ Find this sum.

For a natural number $n$, let $n^{\text{'}}$ denote the number obtained by deleting zero digits, if any.  (For example, if $n=260, n^{\text{'}}=26;$ if $n=2020, n^{\text{'}}=22$). Find the number of $3$-digit numbers $n$ for which $n^{\text{'}}$ is a divisor of $n,$ different from $n.$

Consider the set $E$ of all natural numbers $n$ such that when divided by $11,12,13,$ respectively, the remainders, in that order, are distinct prime numbers in an arithmetic progression. If $N$ is the minimum number in $E$ find the sum of digits of $N$.

Note: In PRMO-2019, original question was asking about maximum number in $E$, which couldn't be calculated. So, this was the bonus question. Only minimum value can be calculated.