Find the smallest square number that is divisible by each of the following numbers: 4, 9, and 10.

Question

Homi · Accepted Answer

We need to find the smallest number that is a perfect square and can be divided by 4, 9, and 10.

Step 1 — Find the Least Common Multiple

First, let us find the Least Common Multiple (LCM) of 4, 9, and 10. The LCM is the smallest number that is a multiple of all these numbers. We find the prime factorization of each number.

$4 = 2 \times 2 = 2^2$

$9 = 3 \times 3 = 3^2$

$10 = 2 \times 5 = 2^1 \times 5^1$

To find the LCM, we take the highest power of each prime factor present in any of the numbers.

$\text{LCM}(4, 9, 10) = 2^2 \times 3^2 \times 5^1$

$= 4 \times 9 \times 5$

$= 36 \times 5$

$\boxed{180}$

Step 2 — Make the LCM a perfect square

A perfect square is a number where all prime factors in its prime factorization have an even power. The LCM we found is 180. Let us look at its prime factorization again.

$180 = 2^2 \times 3^2 \times 5^1$

Here, the prime factor 2 has a power of 2 (even). The prime factor 3 has a power of 2 (even). But the prime factor 5 has a power of 1 (odd).

To make 180 a perfect square, we must multiply it by 5. This will make the power of 5 even ( $5^1 \times 5^1 = 5^2$ ).

$\text{Smallest square number} = 180 \times 5$

$= 900$

We can check its prime factorization:

$900 = 2^2 \times 3^2 \times 5^2$

All prime factors (2, 3, and 5) now have even powers. So, 900 is a perfect square. It is also divisible by 4, 9, and 10 because it is a multiple of their LCM.

Answer

The smallest square number that is divisible by each of the numbers 4, 9, and 10 is 900.

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