Question

Find the five-digit smallest number which can be divided by 15, 55 and 99.

Answer 1

Hint: In this particular question find the LCM of the given numbers (i.e. 15, 55 and 99) and then the smallest five-digit number divisible by 15, 55 and 99 will be equal to the smallest five-digit number divisible by their LCM.

Complete step-by-step answer:
Now first, we had to find the LCM of the number 15, 55 and 99.
Now as we know that LCM of some numbers is the least positive number that is divisible by all the numbers.
And to find the LCM of some numbers first, we had to find the prime factors of all numbers.
So, let us write 15, 55 and 99 into a product of its prime factors.
\[ \Rightarrow 15 = 3 \times 5\]
\[ \Rightarrow 55 = 5 \times 11\]
\[ \Rightarrow 99 = 3 \times 3 \times 11\]
Now the LCM of the numbers will be that number which include all the prime factors of number 15, 55 and 99.
So, LCM of 15, 55 and 99 will be equal to \[3 \times 3 \times 5 \times 11 = 495\]
So, now the smallest five-digit number that is divisible by 15, 55 and 99 will be the smallest five-digit number divisible by their LCM because LCM is already divisible by all the numbers (i.e. 15, 55 and 99).
So, as we know that the smallest five-digit number is 10000.
Now let us divide 10000 by the 495 (i.e. LCM of 15, 55 and 99).
\[495\mathop{\left){\vphantom{1\begin{gathered}
  10000 \\
   - 9900 \\
  {\text{ }}100 \\
\end{gathered} }}\right.
\!\!\!\!\overline{\,\,\,\vphantom 1{\begin{gathered}
  10000 \\
   - 9900 \\
  {\text{ }}100 \\
\end{gathered} }}}
\limits^{\displaystyle \,\,\, {20}}\]
So, the remainder will be 100 when we divide 10000 by 495.
This means that 9900 is divisible by 495. But 9900 is not a five-digit number.
So, the next number which is divisible by 495 will be (9900 + 495) = 10395
So, the smallest five-digit number divisible by 495 (i.e. LCM of 15, 55 and 99) is equal to 10395.
Hence, the smallest five-digit number divisible by 15, 55 and 99 will be 10395.

Note: Whenever we face such a question about the key concept, we have to recall the definition of the LCM of numbers that is “LCM is the smallest number that is divisible by all the numbers”. So, we had to first find the prime factors of the numbers given and then find the set of factors such that it includes all the prime factors of all the numbers and then LCM will be the product of all elements of that set. Now we had to divide the smallest five-digit number (i.e. 10000) by the LCM to find the greatest four-digit