NUMBER OF INTEGERS IN BETWEEN SQUARE OF TWO CONSECUTIVE NUMBERS

Problem 1 :

Number of non perfect square numbers between 12² and 13²

Solution :

Let n = 12² = 144 and (n + 1)² = 13² = 169

2n non perfect squares will be there. Then,

2n = 2 (12)

= 24 non perfect squares

There are 25 non perfect squares are there in between 12² and 13².

Non perfect squares lies between 144 and 169

145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168

Counting it 24 numbers are there.

Problem 2 :

The difference between the squares of two consecutive even numbers is 36 find the numbers

Solution :

The difference between two consecutive numbers = 36

There will be 2n non perfect squares in between two consecutive squares.

2n = 36

n = 36/2

n = 18 and n + 1 = 19

So, the required numbers are 18 and 19.

Problem 3 :

The difference of two positive integers is 5. Two times the square of the smaller integer exceed the square of the larger integer by 71.Find the two integers.

Solution :

The difference between two positive integers = 5

x and x + 5 are the two numbers.

Smaller number = x, larger number = x +  5

2x² = (x + 5)² + 71

2x² = x² + 10x + 25 + 71

2x² - x² - 10x - 96 = 0

x² - 10x - 96 = 0

(x - 16) (x + 6) = 0

x - 16 = 0 and x + 6 = 0

x = 16 and x = -6

Then, the required number is 16. 

x + 15 = 16 + 15 ==> 31

So, the required numbers are 16 and 31.

Problem 4 :

If the sum of two consecutive even numbers plus the next odd number is 29 what was the odd number?

Solution :

Let x be the even number, its consecutive even number will be x + 2. Then x + 3 will be the odd number next to x + 2.

x + x + 2 + x + 3 = 29

3x + 5 = 29

3x = 29 - 5

3x = 24

x = 24/3

x = 8

x + 3 = 8 + 3 ==> 11

So, the required odd number is 11.

Problem 5 :

The sum of three numbers is 87. The third number is twice as large as the second .The second one is one more than the first. Find those numbers.

Solution :

Let x, y and z be the three numbers from smaller to larger.

x + y + z = 87

z = 2y, y = x + 1

x + x + 1 + 2(x + 1) = 87

2x + 1 + 2x + 2 = 87

4x + 3 = 87

4x = 87 - 3

4x = 84

x = 84/4

x = 21

Applying the value of x in y = x + 1, we get

y = 21 + 1

y = 28

applying x and y in x + y + z = 87, we get

21 + 28 + z = 87

z = 87 - 49

z = 38

So, those numbers are 21, 28 and 38.