Question

The least number divisible by 15, 20, 24, 32 and 36 is ?
A. 1440
B. 1660
C. 2880
D. None of these

Answer 1

Hint: Here we find the least number which is divisible by these 5 numbers by calculating the LCM of the 5 numbers. We write each number in the form of its prime factors and write the LCM of the numbers as multiplication of prime factors with their highest powers.
* Prime factorization is a process of writing a number in terms of its prime factors.
* We use the concept \[\underbrace {a \times a..... \times a}_n = {a^n}\]

Complete step-by-step answer:
We are given the numbers 15, 20, 24, 32 and 36.
We write the prime factorization of each number.
\[15 = 3 \times 5\]
\[20 = 2 \times 2 \times 5\]
\[24 = 2 \times 2 \times 2 \times 3\]
\[32 = 2 \times 2 \times 2 \times 2 \times 2\]
\[36 = 2 \times 2 \times 3 \times 3\]
Now use the concept of \[\underbrace {a \times a..... \times a}_n = {a^n}\]to write the prime factorization in a simple form where n represents the power of the prime factors.
\[15 = 3 \times 5\]
\[20 = {2^2} \times 5\]
\[24 = {2^3} \times 3\]
\[32 = {2^5}\]
\[36 = {2^2} \times {3^2}\]
Now we calculate the LCM of the numbers by multiplying the maximum degree terms of prime factors.
Since, the highest power of 2 is 5, highest power of 3 is 2 and highest power of 5 is 1.
\[ \Rightarrow \]LCM \[ = {2^5} \times {3^2} \times 5\]
Calculate the value of RHS of the equation.
\[ \Rightarrow \]LCM \[ = 32 \times 9 \times 5\]
\[ \Rightarrow \]LCM \[ = 1440\]

So, option A is correct.

Note: Students might get confused between the concept of highest common factor (HCF) and Lowest common multiple (LCM). LCM gives us the lowest common multiple of various numbers and HCF gives us the highest common divisor of the numbers. Keep in mind that while calculating the LCM we have to cover all the factors of the number and in HCF we multiply only the common prime factors having the lowest power.
Students many times write the highest power of a prime factor and then again write that prime factor thinking that it is occurring in the prime factorization of another number which is wrong, once you’ve taken the highest power, which means you have covered all possible factors of that prime number.