Question

The least perfect square number which is divisible by 8, 15, 20, 22 is
A. 435600
B. 43560
C. 39600
D. None

Answer 1

Hint: LCM (Least Common Multiple) of a set of numbers is the least number which is a multiple of all of them, or which is divisible by all the numbers of the set.
Any multiple of LCM will also be divisible by the numbers of the set, but it would no longer be the least.
A perfect square can be written as a product of two identical numbers.

Complete step-by-step answer:
Let us first find the LCM of the given numbers by factoring them into prime numbers:
$8={{2}^{3}}$
 $15=3\times 5$
$22=2\times 11$
 $20={{2}^{2}}\times 5$
Since an LCM is a multiple of all of them, it must have at least the maximum power of all the prime numbers present in it.
∴ The LCM will be ${{2}^{3}}\times 3\times 5\times 11$ .
Now we want a number which should be divisible by all the given numbers, so it must be a multiple of the LCM.
Since, the number we want must also be a perfect square, we need to find a multiple of the LCM which can be written as a product of two identical numbers.
But a prime number cannot be divided into parts, so we must multiply the LCM with a number so that every power of the prime numbers in the group becomes an even power (can be separated into two identical parts).
The least such perfect square will be obtained if we multiply the LCM by 2 × 3 × 5 × 11 to give ${{2}^{4}}\times {{3}^{2}}\times {{5}^{2}}\times {{11}^{2}}$ .
We can calculate its value as well: 16 × 9 × 25 × 121 = 453600.
∴ The answer is A. 453600.

Note: There are infinitely many perfect square common multiples, but the least one is obtained by multiplying the LCM with the least such number that gives even powers for all its prime factors.
Similarly, we can find least cube common factors or other higher-powered common factors.
Prime Numbers: The numbers which have exactly two factors, i.e. they are not divisible by any other number except for 1 and themselves, are called prime numbers. All other numbers are called composite numbers. 1 is neither prime (it has exactly one factor!), nor composite.