Question

Find the smallest number which, on being decreased by $3$, is completely divisible by $18, 36, 32$ and $27$.

Answer 1

Hint: To find the smallest number that is decreasing by $3$ divides $18, 36, 32$ and $27$ completely, we need to find the LCM of $18, 36, 32$ and $27$ and then add 3 to the LCM that we will obtain. To find the LCM of these numbers, we will use the prime factorization method.

Complete step by step solution:
In this question, we are given 4 numbers and we need to find out the smallest number which when decreased by 3 is divisible by these numbers.
So, the smallest number that divides all these 4 numbers will be the LCM of these numbers. So, here we have to find the LCM of 18, 36, 32 and 27 to find this smallest number.
Steps for finding LCM using Prime factorization:
Step 1:
Prime factorize all the given numbers.
$ \Rightarrow $Prime Factors of 18$ = 2 \times 3 \times 3 = 2 \times {3^2}$
$ \Rightarrow $Prime factors of 36$ = 2 \times 2 \times 3 \times 3 = {2^2} \times {3^2}$
$ \Rightarrow $Prime factors of 32$ = 2 \times 2 \times 2 \times 2 \times 2 = {2^5}$
$ \Rightarrow $Prime factors of 27$ = 3 \times 3 \times 3 = {3^3}$

Step 2:
Now, the LCM of these four numbers will be equal to the product of the highest power of the prime numbers. Therefore,
$ \Rightarrow $LCM of 18,36,32,27$ = {2^5} \times {3^3} = 32 \times 27 = 864$
But, we need to find the number that is being decreased by 3. Therefore, our answer will be
$ \Rightarrow 864 + 3 = 867$
Hence, if we decrease 867 by 3, 864 will be divisible by all 18, 36, 32 and 27.

Note:
Here the most important part in this question is that we are finding the LCM. As the smallest number is completely divisible by these 4 numbers, that means we have to find the least common multiple.