finding the least square number divisibe by given numbers

We are required to find the least possibe square number that is exactly divisible by the numbers 10, 12 and 16. In this question, I found out the required least possible square number is 3600. I did that by hit and trial method by taking out number randomly and then squaring and checking to see if the square number is exactly divided by a the three numbers. I just want to know the definite way to find the answer if there is any.

• $900$ isn't divisible by $16$. – Daniel Fischer Jun 2 '14 at 14:01
• Look at the prime factorizations of $10$, $12$, $16$. The answer is in there. – Jyrki Lahtonen Jun 2 '14 at 14:02

2 Answers

$10=2\times 5$

$12=2^2\times 3$

$16=2^4$

From here, we obtain $\text{lcm} (10,12,16)=2^4\times 3\times 5$, hence the least square number required is $2^4\times 3^2\times 5^2=3600$.

• Nice, the smallest number divisible by any quantity of numbers is the lcm of those numbers, and then you just made all the exponents even so it was a square right? – Jorge Fernández Hidalgo Jun 2 '14 at 14:04
• taking number for exampe 10,12 and 18, the lcm of those numbers is 180 and squaring 180 makes it 32400. However, 900 is also a square number and is divisible by a three numbers. – Ufomammut Jun 3 '14 at 0:14
• Newer question asked by OP to clarify here. – 6005 Jun 4 '14 at 3:00

The key observation is that if a prime $p$ divides $n^2$ then $p$ divides $n$.

Now consider the given numbers:

$10 = 2 \cdot 5$

$12 = 2^2 \cdot 3$

$16 = 2^4$

This means that you need $n$ to be a multiple of $2^2 \cdot 3 \cdot 5$.

The smallest such $n$ is $60$ and square you seek is $3600$.