Question

How many numbers from 1 to 1000 are exactly divisible by 60 but not 24 is? \[\]

Answer 1

Hint: We use the formula number of multiples of a number $p$ from 1 to $n$ as $\left[ \dfrac{n}{p} \right]$ where $\left[ \cdot \right]$ is greatest integer function to find the number of multiples of 60 from 1 to 1000. We find the number of common multiples of 24 and 60 from 1 to 1000 by finding the number of multiples of their least common multiple(LCM) from 1 to 1000. We subtract the number of common multiples from the number of multiples of 60 to get the answer. \[\]

Complete step by step answer:
We know the greatest integer function any real number $x$ returns the greatest integer less than equal to$x$. It means if $x=2.00001$ then $\left[ x \right]=2$ since 2 is greatest integer less than equal to 2.00001. \[\]
We have the multiples of 60 from 1 to 1000 are 60, 120, 180 and so on. The number of multiples of 60 from 1 to 1000 will be the great integer less than equal to the quotient obtained in decimal division of 60 by 1000 which $\left[ \dfrac{1000}{60} \right]=\left[ 16.\overline{6} \right]=16$\[\]
Now we find the number of common multiples of 60 and 24. The common multiples of 60 and 24 will also be the multiples of their least common multiple (LCM). Let us find the LCM of 60 and 24. We have their prime factorisation as
\[\begin{align}
  & 60=2\times 2\times 3\times 5 \\
 & 24=2\times 2\times 2\times 3 \\
\end{align}\]
We take product of primes where highest number of times the prime is multiple with itself. We take $2\times 2\times 2$ from 24 and $3\times 5$from 60 and have the LCM as
\[\operatorname{LCM}\left( 60,24 \right)=2\times 2\times 2\times 3\times 5=120\]
The number of multiples of 120 from 1 to 1000 is equal to number of common multiples of 60 and 24 from 1 to 1000 that is $\left[ \dfrac{1000}{120} \right]=\left[ 8.\overline{3} \right]=8$
So we have 16 multiples of 60 out of which 8 are also multiples of 24. So the number of multiples of 60 but not 24 is $16-8=8$.\[\]

Note:
We note that when the dividend is exactly divisible by the divisor then we call the divisor a factor of dividend and the dividend a multiple of divisor. The prime factorization of number $N$ is given as $N={{p}_{1}}^{{{e}_{1}}}{{p}_{_{2}}}^{{{e}_{2}}}...{{p}_{n}}^{{{e}_{n}}}$ with its prime factors ${{p}_{1}},{{p}_{2}},...,{{p}_{n}}$ and integral exponents${{e}_{1}},{{e}_{2}},...,{{e}_{n}}$. The greatest integer function is also written as $GIF\left( x \right)$ and also called floor function.