Question

18 is divisible by both 2 and 3. It is also divisible by \[2\times 3=6\]. Similarly, a number is divisible by 4 and 6. Can we say that the number must be divisible by \[4\times 6=24\]? If not, give an example to justify your answer.

Answer 1

Hint: In this type of question we have to use the concept of divisibility. Divisible means “when we divide a number by another number the result is a whole number and the remainder is equal to zero”. We know that if a number is divisible by two co-prime numbers, then it is divisible by their product also. Also we know that a number is said to be prime if it is divisible by 1 and the number itself.

Complete step-by-step solution:
Now we have given that, 18 is divisible by both 2 and 3, It is also divisible by \[2\times 3=6\]. And we have to check if a number is divisible by 4 and 6 then it is divisible by \[4\times 6=24\] with the help of an example.
As we know, if a number is divisible by two co-prime numbers, then it is divisible by their product also. Here, 2 and 3 are co-prime numbers and hence, if 18 is divisible by both 2 and 3 it is also divisible by \[2\times 3=6\].
Now, we can see that in case of 4 and 6, 4 and 6 are not coprime numbers, and hence the number divisible by 4 and 6 will not be divisible by \[4\times 6=24\].
For example, let us consider the numbers which are divisible by 4
\[\Rightarrow 4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,\cdots \cdots \cdots \]
Now, let us consider the numbers which are divisible by 6
\[\Rightarrow 6,12,18,24,30,36,42,48,54,60,\cdots \cdots \cdots \]
So that we get the numbers which are divisible by both 4 and 6 are
\[\Rightarrow 12,24,36,48,60,\cdots \cdots \cdots \]
Here, we can observe that the numbers \[12,36,60,\cdots \cdots \cdots \] are divisible by both 4 and 6 but not divisible by 24.
Hence, we can conclude that a number is divisible by 4 and 6 need not be divisible by 24.

Note: In this type of question students have to remember that if a number is divisible by two co-prime numbers, then it is divisible by their product also and they have to note that it is applicable only for co-prime numbers. Also students have to learn the divisibility rules for 1 to 20, which will help them to make division procedure easier.