The smallest square number which is exactly divisible by each of the numbers 6,9 and 15 is _________. A) 100 B) 400 C) 900 D) 1024
Hint: We have to find the smallest square number which is exactly divisible by each of the given numbers so that we have taken LCM by using the prime factorization method. And we have to make LCM a perfect square by multiplying with a positive real number.
Complete step-by-step answer: As we know that we have to find the smallest square number which is exactly divisible by each of the numbers 6,9 and 15 So we take LCM (LOWEST COMMON FACTOR) by prime factorization method. So LCM of 6,9,15 \[6 = 2 \times 3\] $9 = 3 \times 3$ $15 = 3 \times 5$ From this LCM of 6,9,15 = $2 \times 3 \times 3 \times 5$ LCM of 6,9,15 = $90$ So we have $2 \times 3 \times 3 \times 5$ but this is not a perfect square To make $2 \times 3 \times 3 \times 5$ as perfect square this we have to make equal pair of 2,3,5 as we see that 3 is in pair but 2 and 5 is not in pair so to making pair we have to multiply it by 2,5 so that it is perfect square. So we multiply it by 2 and 5 we get $2 \times 2 \times 3 \times 3 \times 5 \times 5$ So the required square number is $900$ So option C is the correct answer.
Note: An important point of this question is making squares by multiplying numbers. If same question say about smallest cube than all the step are same but LCM that we find, we have to convert this in cube for cube we have to make pair of three same digit like $4 \times 4 \times 4$. We can find LCM by using a division method so in this we have to multiply all the divisors to find LCM.