Question

Aqua
3.
Find the smallest perfect square that is exactly divisible by each of the following numbers
(a) 8, 9 and 10
(b) 8. 15 and 20
(c) 6. 9 and 15
4. Check by prime factorisation method, which of the followi
are perfect squares?​

Answer 1

Answer:

(a) 3600

(b) 3600

(c) 900

Step-by-step explanation:

We know that,

The smallest perfect square must be divisible by all the numbers given,

So, for that to happen, it must be a multiple of the given numbers (Think about it.....)

Only if a number is a multiple of another number can it be divisible,

For ex:- 8 is a multiple of 4

Thus, 8 is divisible by 4, 8 ÷ 4 = 2

Now, back to our Question,

(a)

Thus, to find a number which is divisible by 8, 9 and 10

We must find its LCM of 8, 9 and 10

LCM = 360

Now, if we take the square root of 360 we will not get a perfect square because its prime factorization in not in pairs,

360 = 2 × 2 × 2 × 3 × 3 × 5

Here 2 and 5 is not in a pair so we must multiply 2 and 5 with 360,

360 × 2 × 5

= 360 × 10

= 3600

Now, it is a Perfect square because,

3600 = 2 × 2 × 2 × 2 × 3 × 3 × 5 × 5

Since all numbers are in pair and it is divisible by 8, 9 and 10.

Thus, the smallest perfect square that is exactly divisible by 8, 9 and 10 is 3600

(b)

Similarly,

LCM of 8, 15 and 20 = 120

Now,

120 = 2 × 2 × 2 × 3 × 5

Here 2, 3 and 5 is not in pairs so 120 is not a perfect square,

thus, we must make it by multiply 2, 3 and 5 to 120.

= 120 × 2 × 5 × 3

= 120 × 10 × 3

= 360 × 10

= 3600

Thus, the smallest perfect square that is exactly divisible by 8, 15 and 20 is 3600.

(c)

Here also same process,

LCM of 6, 9 and 15 = 90

Now,

90 = 2 × 3 × 3 × 5

Again 90 is not a perfect square because its 2 and 5 are not in pairs so we multiply 2 and 5 to 90

= 90 × 2 × 5

= 90 × 10

= 900

Thus, the smallest perfect square that is exactly divisible by 6, 9 and 15 is 900

Hope it helped and you understood it........All the best