- How many numbers are there less than 100 that cannot be written as a multiple of a perfect square greater than 1?
- Find the smallest number that has exactly 18 factors?

Qn1 :How many numbers are there less than 100 that cannot be written as a multiple of a perfect square?

To begin with, all prime numbers will be part of this list. There are 25 primes less than 100. (That is a nugget that can come in handy)

Apart from this, any number that can be written as a product of two or more primes will be there on this list. That is, any number of the form pq, or pqr, or pqrs will be there on this list (where p, q, r, s are primes). A number of the form p^{n}q cannot be a part of this list if n is greater than 1, as then the number will be a multiple of p^{2}.

This is a brute-force question.

Post this, we need to count all numbers of the form p*q*r, where p, q, r are all prime.

There is another method of solving this question.

We can list all multiples of perfect squares (without repeating any number) and subtract this from 99

16 – 0 new multiples

25 – 3 new multiples { 25, 50, 75

36 – 0 new ones

49 – 2 { 49, 98}

64 – 0

81 – 0

So, total multiples of perfect squares are 38. There are 99 numbers totally. So, there are 61 numbers that are not multiples of perfect squares

This is a difficult and time-consuming question. But a question that once solved, helps practice brute-force counting. Another takeaway is the fact that there are 25 primes less than 100. There is a function called pi(x) that gives the number of primes less than or equal to x. pi(10) = 4, pi(100) = 25

Qn2: Find the smallest number that has exactly 18 factors?Any number of the form p^{a}q^{b}r^{c} will have (a+1) (b+1)(c+1) factors, where p, q, r are prime. (This is a very important idea)

^{a}q

^{b}, then it can be of the form

^{1}q

^{8}or p

^{2}q

^{5}

^{a}, then it can be of the form

^{17}

^{a}q

^{b}r

^{c}, then it can be of the form

^{1}q

^{1}r

^{2}

^{17}, p

^{2}q

^{5}, p

^{1}q

^{8}or p

^{1}q

^{2}r

^{2}

**prime factorizations that can result in a number having 18 factors.**

__only possible__^{17}– Smallest number = 2

^{17}

^{2}q

^{5 }– 3

^{2}* 2

^{5}

^{1}q

^{8}– 3

^{1}* 2

^{8}

^{1}q

^{1}r

^{2}– 5

^{1}* 3

^{1}* 2

^{2}

^{1}* 3

^{1}* 2

^{2}= 180