Question

What is a multiple of 2 and 7?

Answer 1

Hint: We know that the LCM of two numbers is the smallest common multiple. So, we can say that the multiples of these will be LCM, the common multiples of those numbers. So, we can find the LCM of 2 and 7 and any one of the multiples of this LCM will be our required answer.

Complete step by step solution:
We know that if a number a is perfectly divisible by another number b, then we say that the number a is the multiple of number b.
Here, in this question, we need to find a common multiple for the two given numbers 2 and 7.
We know that for two numbers, a and b, the smallest number that is divisible by both of these numbers, is called the Lowest Common Multiple or the LCM of those two numbers.
So, if we try to list the multiples of this LCM, we will get the common multiples of those two numbers.
So, let us first find the LCM of 2 and 7.
We know that the prime factorisation of 2 is
$\begin{align}
  & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
$\therefore 2=2\times 1$
We know that the prime factorization of 7 is
$\begin{align}
  & 7\left| \!{\underline {\,
  7 \,}} \right. \\
 & \text{ }\left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
$\therefore 7=7\times 1$
So, we can clearly see that the maximum number of times of occurrence of 2 = 1 …(i)
And, the maximum number of times of occurrence of 7 = 1 …(ii)
Hence, by using equations (i) and (ii), we can say that 2 must occur once, and 7 must also occur once. Thus, we can say that
$LCM=2\times 7$
$\Rightarrow LCM=14$
Hence, the LCM of 2 and 7 is 14.
As per the above discussion, we can say that all the multiples of 14 will be a multiple of 2 and 7.
Hence, we can say that any multiple of 14, such as 14, 28, 42, 56, … , will be a multiple of both 2 and 7.

Note: We just need to find one multiple, so the easiest solution will be to multiply these two numbers, and the product will be our required answer. We must also remember that the LCM of two primes is always the product of those two prime numbers.