Question

Find the least number that can be divided exactly by all the even numbers between $ 10{\text{ and 20}} $

Answer 1

Hint: To find the least number that can be divided exactly, first we will find the even numbers between $ 10{\text{ and 20}} $ . Then, by using the concept of LCM (Least common multiple) applicable to get the number exactly, and for LCM we need to find the prime factors first.

Complete step-by-step answer:
We know that even numbers between $ 10{\text{ and 20}} $ are $ 12,14,16{\text{ and 18}} $
Prime factorization is the process of finding which prime numbers can be multiplied together to make the original number.
Prime factors of –
\[
\Rightarrow 12 = \underline 2 \times 2 \times 3 \\
\Rightarrow 14 = \underline 2 \times 7 \\
\Rightarrow 16 = \underline 2 \times 2 \times 2 \times 2 \\
\Rightarrow 18 = \underline 2 \times 3 \times 3 \\
 \]
Now, LCM of the given numbers is writing common factors into the product of other factors.
Therefore, LCM $ = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 7 $
LCM $ = 1008 $
Hence, the least number exactly divisible by the even numbers between $ 10{\text{ and 20}} $ is $ 1008 $

Note: Since, the given numbers were very small we have written its factors directly, you can use the Division method or the prime factor tree to get prime factors in case of large numbers. To solve these types of sums, one should be clear about the concept of HCF and LCM and the prime numbers. HCF is the highest or greatest common multiple whereas the LCM is the least common multiple or least common divisor in two or more given numbers. Prime numbers are the numbers greater than $ 1 $ and which are not the product of any two smaller natural numbers. For Example: $ 2,{\text{ 3, 5, 7,}}...... $ $ 2 $ is the prime number as it can have only $ 1 $ factor. Factors are the number $ 1 $ and the number itself. Also, remember that we get the prime factorization of any composite number.