Question

What is the smallest 5 digit number that is exactly divisible by 72 and 108?

Answer 1

Hint: In this particular question use the concept that the least number by which the given numbers is exactly divisible is the LCM of the numbers so first of all calculate all the prime factors of the given number for example the prime factors of 10 are (1, 2 and 5), prime factors are those which is divide by 1 or itself so use these concepts to reach the solution of the question.

Complete step-by-step answer:
Given numbers are 72 and 108.
We have to find out the L.C.M of these numbers.
L.C.M is called the least common multiple.
The least common multiple (LCM) of any two numbers or more than two numbers is calculated as first list all the prime factors of each number, then multiply each factor the greatest number of times it occurs in the numbers. If the same factor occurs more than once in the given numbers then we have to multiply the factor the greatest number of times it occurs.
Therefore, first calculate the factors of the first number i.e. 72. So, the factors of 72 are,
$72 = 1 \times 2 \times 2 \times 2 \times 3 \times 3$
Now calculate the factors of the second number i.e. 108. So, the factors of 108 are
$108 = 1 \times 2 \times 2 \times 3 \times 3 \times 3$
The L.C.M of the given numbers according to above definition is calculated as
L.C.M = $1 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3$ = 216
So the L.C.M of 72 and 108 is $= 216$
Now we have to find the smallest 5 digit number that is divisible by that is exactly divisible by 72 and 108.
So the smallest 5 digit number should be the multiple of LCM of $72$ and $108$ (i.e. 216).
Now as we know that the smallest five digit number is $10000$.
Now divide this number by $216$ we have,
$ \Rightarrow \dfrac{{10000}}{{216}} = 46\dfrac{{64}}{{216}}$
So as we see that the quotient is $46$ and remainder is $64$.
So the number which is exactly divisible by $216$ is $(10000 – 64) = 9936$.
Now in this number add $216$, which is the required smallest $5$ digit number which is exactly divisible by $72$ and $108$.
So the smallest $5$ digit number is $= 9936 + 216 = 10152$.
Therefore, the smallest 5 digit number that is exactly divisible by $72$ and $108$ is $10152$.

Note: Whenever we face such types of questions first find out the LCM of the numbers then write the smallest 5 digit number (i.e. 10000) then divide this number by 216 and note down the remainder, then subtract this remainder from the smallest 5 digit number then add the LCM value in the resultant number of the given numbers and simplify which is the required answer.