Question

Find the numbers between \[1\] and $100$ which are divisible by $2$ and $5$.

Answer 1

Hint:We need to consider the factors of the given numbers as well as the LCM of the given number. We know that if a number divides another number then every factor of that number divides the bigger number as well. So, we will use this fact and find the numbers which satisfy the given criteria.


Complete step-by-step answer:
Let $a$ and $b$ be two numbers and $l$ be the LCM of the numbers $a$ and $b$ then if $l$ divides a number $d$ then the numbers $a$ and $b$ both divide the number $d$ .
The given numbers are $2$ and $5$ .
So, we will first calculate the LCM of these two numbers.
Observe that both the numbers are co-prime to each other.
Therefore, the LCM of both the numbers is just the product of two numbers.
Therefore, the LCM of the given numbers is calculated as follows:
$2 \times 5 = 10$
Therefore, all the multiples of $10$ between $1$ and $100$ will have the factor $2$ as well as $5$.
Now, the multiples of $10$ are $10,20,30,...,90$.
Therefore, all the numbers in the above list are divisible by $10$ .
Thus we can say that the numbers which are divisible by $10$ are also divisible by $2$ and $5$.

Thus, the correct answer is all the multiples of number $10$ .

Note:Observe that we first considered a single factor and found out all the numbers which are divisible by both the numbers and the LCM of the numbers. So first we will calculate the LCM of the given numbers and then consider multiples of the LCM.