The least perfect square, which is divisible by each of 21,36 and 66 is (options)

Question

(a) 213444
(b) 214344
(c) 214434
(d) 231444

Any short method to solve this question in 1 min?

Answer 1

HINT: Find LCM of $M=21,36,66$

$(1)$

Check for the prime factorization of $M$

If prime $p$ has odd index, multiply $M$ by $p$

Or $(2)$

Check for the divisibility of the given numbers by $M$

Answer 2

Only one of them is divisible by 7, so it has to be the answer. If you can do divisibility checks for these numbers in less than 1 min, that would suffice.

For example, you can do something like this rather quickly in your head: $$ 214434 = 210000 + 4434 \equiv 4434 = 4200 + 234 \equiv 234 = 210 + 24 \equiv 24 \not\equiv 0 \quad(\text{mod }7) $$

Of course, if each number turned out to be divisible by e.g. 6, 7, 11 and other small factors of $\mathrm{lcm}(21,36,66)$, this method wouldn't be very fast.

Answer 3

LCM of 21, 36, 66 is 3^2, 2^2, 7, 11. So the smallest perfect square should be 3^2 x 2^2 x 7^2 x 11^2 = 213444.