Find the smallest cube number which is divisible by 72, 108 and 300?

Solution:

Find LCM of 72, 108, 300

Prime factors:
72 = $2^3 \times 3^2$
108 = $2^2 \times 3^3$
300 = $2^2 \times 3 \times 5^2$

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

To make it a perfect cube, powers must be multiples of 3:

• $2^3$ ✔️
• $3^3$ ✔️
• $5^2$ ❌ → need one more 5

So multiply by 5:

$= 2^3 \times 3^3 \times 5^3 = (2 \times 3 \times 5)^3 = 30^3$

$30^3 = 27000$

✅ Final Answer: 27000