What is the smallest square number which is divisible by 4, 6 and 32?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 576
A number divisible by 4, 6 and 32 is a multiple of their LCM. A perfect square has every prime factor an even number of times, so the LCM is multiplied by the primes it is short of.
Prime factors: \(4=2^2\), \(6=2\times3\), \(32=2^5\)
LCM, the highest power of each prime: \(2^5\times3=96\)
2 appears 5 times and 3 once; one more 2 and one more 3 make both counts even: \(96\times2\times3=576\)
So the square is \(576=2^6\times3^2=24^2\)
Shortcut:
Only 96 and 576 among the options are divisible by 32.
96 is not a perfect square, while \(576=24^2\).