Question

Find a number which has a multiple of all the numbers from 1 to 10?
A. 5040
B. 1260
C. 720
D. 1440

Answer 1

Hint:In the problem they have asked us to find the number which has a multiple of all the numbers from 1 to 10, we go for the least common multiple (LCM) of all the numbers from 1 to 10. Implies the least number which is a multiple of 1, 2, 3,…, 10. Either the obtained LCM or its multiple will be the choice we need.


Complete step-by-step answer:
Least common multiple (LCM) of numbers a, b is defined to be the least number which is a multiple of both a and b.
We need a number from the choices which has a multiple of all the numbers from 1 to 10 implying that the required answer should have the factors 1, 2, 3, …, 10. Thus let us check, which is the least common multiple (LCM) of all the numbers from 1 to 10. The LCM we require will be nothing but the value after multiplying all numbers from 1 to 10.
LCM (1, 2, 3, 4, …, 10) $=2520$
Either the above LCM will be the solution as its factors include all the numbers from 1 to 10, or its multiple from the choices.
As we have $5040=2520\times 2$ we have 5040 as the multiple of 2520.
5040 is the number having multiple of all the numbers from 1 to 10.

So, the correct answer is “Option A”.

Note:LCM (1,2,3, …, 10)= the least number which is a multiple of all the numbers from 1 to 10. Number 7 is the one which is not a factor or multiple of any other numbers from 1 to 10. Thus choose 7. Now as we have 2 as a factor of 4,6 and 10, we eliminate the common factor 2. And $8=2\times 4$, $9=3\times 3$ and $10=2\times 5$, we eliminate 8, 9, 10. Thus choose 3, 4, 5, 6, 7 which are the only numbers required to find the LCM from 1 to 10. Thus we obtain $3\times 4\times 5\times 6\times 7=2520$.