Question

1008 - By which smallest number the given number should be multiplied,so that the result becomes a perfect square ? Find the square root of the result obtained.

Answer 1

any number will be perfect square only when unit digit of number contains 1, 4, 5, 6, or 9.

so, we have to choose a smallest number ' a ' in such a way that unit digit of 1008a = 1, 4, 5, 6 or 9.

if we take a = 1
then, 1008 × 1 => unit digit 8 so, can't possible
if we take a = 2
then, 1008 × 2 = 2016 => unit digit of 6, may be possible. Let's check .
45² = 2025 and 44² = 1936
1936 < 2016 < 2025 , 44² < 2016 < 45²
so, 2016 ≠ a perfect square.

if we take a = 3 ,
then, 1008 × 3 = 3024 unit digit is 4 so, may be possible. but we see 3025 = 55² so, 3024 ≠ a perfect square.

if we take a = 4
then, 1008 × 4 = 4032 => not possible

if we take a = 5
then, 1008 × 5 = 4040 => not possible

if we take a = 6
then 1008 × 6 = 6048 => not possible

if we take a = 7
then, 1008 × 7 = 7056 => unit digit 6 , may be possible.
7056 = 84 × 84 = 84²
hence, if we take a = 7 then number 1008a will be perfect square.

hence smallest number is 7 and square root of number is 84.

Answer 2

Since it is a 4 digit number so, the square root must be a 2 digit number.

By hit and trial method we have to find two perfect squares in both upper and lower bound to verify which interval is shortest.

So, we know that 32² = 1024 which is greater than 1008.

So to make 1008 equal to 1024 we have to multiply (1024 / 1008) = 1.015

But, this case was valid for float and not for integers.

So, the method we need to apply is find out factors of 1008.

1008 = 2*2*2*2*3*3*7

So, from these factors let's make pair: (2*2) * (2*2) * (3*3) * (7)

Since 7 doesn't have a pair, so by multiplying 7 we can make 1008 a perfect square.

The perfect square number will be 7056.