Question

If a number is divisible by 2 and 3, then it satisfies the divisibility rule of
(a) 5
(b) 6
(c) 4
(d) 7

Answer 1

Hint: To solve these types of question we will assume a variable which is divisible by both 2 and 3 and then through that number, we will check the divisibility of another number.

Complete step-by-step solution -
In the question given, we have to find the number(s) by which any number will be divisible when the number is divisible by both 2 and 3. To do this, we assume that a variable is divisible by both 2 and 3. Let this variable be ‘p’. We know that if the number p is divisible by 2, then the least power of 2 in that number will be equal to 1. Similarly, if the number p is divisible by 3, then the least power of 3 in that number will be equal to 1. Thus, we can say that p will be multiplication of 2 and 3 and a constant. Thus, we get
p = 2 x 3 x k ……………….(i)
where k can have any integer value. On simplifying the equation (i) we get,
p = 6k
On rearranging terms, we get,
$\dfrac{p}{6}=k$
This means that when we will divide p by 6, we will get a constant. This k will be the quotient of division. Now, we know that if quotient ‘k’ is a natural number and upon division, if it leaves the remainder 0 then we will say that p is divisible by 6. Thus, we can say a number divisible by 2 and 3 both will also be divisible by 6.
Hence option (b) is correct.

Note: The alternate method to solve this question is by finding the LCM of the two numbers which are divisible. The LCM will also be divisible by 6. In the above case, the LCM of 2 and 3 is 6. Hence the number is divisible by 6.