HOW TO FIND THE LEAST NUMBER TO BE SUBTRACTED TO GET A PERFECT SQUARE

Problem 1 :

What is the smallest number to be subtracted from 549162 in order to make it a perfect square?

a)   28     b)   36     c)   62     d)   81

Solution :

Put comma for every two digits starting from the last.

To get the next step, we will multiply the quotient by 2.

Now multiply 74 by 2, we get 148.

The nearest perfect square (741)²

(741)² = 549081

549081 < 549162

To get the number to be subtracted = 549162 - 549081

= 81

So option (d) is correct.

Problem 2 :

What is the least number which should be subtracted from 0.000326 to make it a perfect square?

a)   0.000002   b)   0.000004    c)   0.02    d)   0.04

Solution :

The given number is 0.000326 = 326 × 10^-6

= (324 + 2) × 10^-6

= (18 × 18 + 2) × 10^-6

326 - 2 is a square number.

Therefore, 2 × 10^-6  = 0.000002 should be subtracted from 0.000326.

So, option (a) is correct.

Problem 3 :

A general wishes to draw up his 36581 soldiers in the form of a solid square. After arranging them, he found that some of them are left over. How many are left ?

(a)  65     (b) 81   (c)  100   (d) None

Solution :

So, 100 is the left over.

Problem 4 :

Find the smallest number that must be subtracted to 1780 to make it a perfect square.

Solution :

(42)² = 1764

1764 < 1780

So, the required number to be subtracted is 16.

Problem 5 :

Find the least number which must be subtracted from 3250 so as to get a perfect square

Solution :

So, 1 to be subtracted to make 3250 as perfect square.

Problem 6 :

What is the least number should be subtracted from 1385 to get a perfect square ? Also find the square root of the perfect square.

Solution :

From the calculation above, 16 is the number to be subtracted from 1385 to make it as perfect square.

Problem 7 :

Find the least number which must be subtracted from each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.

(a) 402     (b) 1989    (c) 3250    (d) 825

Solution :

a) 402

Nearest perfect square for 402 is 400. Then this 2 is extra. So, 2 is the number to be subtracted from 402 to make it as perfect square.

√400 = √(20 x 20)

= 20

So, 2 is the number to be subtracted from 402 to make it as perfect square. The square root of the number obtained after subtracting, we get 20.

b) 1989

This 53 is extra. Then by subtracting 53 from 1989, we get

1989 - 53 = 1936

√1936 = √(44 x 44)

= 44

So, 53 is the number to be subtracted from 1989 to make it as perfect square. The square root of the number obtained after subtracting, we get 44.

c) 3250

This 1 is extra. Then by subtracting 1 from 3250, we get

3250 - 1 = 3249

√3249 = √(57 x 57)

= 57

So, 1 is the number to be subtracted from 3250 to make it as perfect square. The square root of the number obtained after subtracting, we get 57.

d) 825

This 41 is extra. Then by subtracting 41 from 825, we get

825 - 41 = 784

√784 = √(2 x 7 x 2 x 7 x 2 x 2)

= 2 x 2 x 7

= 28

So, 41 is the number to be subtracted from 825 to make it as perfect square. The square root of the number obtained after subtracting, we get 28.

Problem 8 :

Find the least number which must be added to each of the following numbers so as to get a perfect square. Also find the square root of the perfect square so obtained.

(a) 525        (b) 1750      (c) 1825

Solution :

(a) 525

22² = 484

23² = 529

529 - 525 ==> 4

So, 4 is the number to be added to make it as perfect square.

√529 = √(23 x 23)

= 23

After adding 4, we get the answer 23.

(b) 1750 

41² = 1681

42² = 1764

1764 - 1750 ==> 14

1750 + 14 ==> 1764

So, 14 is the number to be added to make it as perfect square.

√1764 = √(2 x 2 x 3 x 7 x 3 x 7)

= 2 x 3 x 7

= 42

After adding 14, we get the answer 42

(c) 1825 

42² = 1764

43² = 1849

1849 - 1825 ==> 24

1825 + 24 ==> 1849

So, 24 is the number to be added to make it as perfect square.

√1849 = √(43 x 43)

= 43

After adding 24, we get the answer 43