NCERT Exemplar Solutions
Class 6 - Mathematics - Unit 1: Number System - True or False Questions
Question 81

Question. 81

If a number is divisible both by 2 and 3, then it is divisible by 12.

false

Suppose the number is n. Given: \(2 \mid n\) and \(3 \mid n\) (that is, 2 divides n and 3 divides n).
From this we can say \(6 \mid n\), because the LCM of 2 and 3 is 6.
To be divisible by 12, we need \(12 \mid n\). In prime factors, \(12 = 2^2 \times 3\).
So, for \(12 \mid n\), the number must have two factors of 2 and one factor of 3.
But from \(2 \mid n\) and \(3 \mid n\), we only know there is at least one 2 and one 3 in n. We are not sure about the extra 2.
Counterexample: take \(n = 18\).
Check: \(18 \div 2 = 9\) (divisible by 2) and \(18 \div 3 = 6\) (divisible by 3).
But \(18 \div 12 = 1.5\) (not a whole number), so \(12 \nmid 18\).
Therefore, a number divisible by 2 and 3 is divisible by 6, but not necessarily by 12. Hence, the statement is false.