Rational Numbers | FIO

Question 7

How many numbers lie between the squares of the following numbers?

(i) 16 and 17 (ii) 99 and 100

Solution

Understand the Question

To find the count of non-perfect square numbers lying strictly between the squares of two consecutive natural numbers $n$ and $n + 1$ , we count the numbers from $n^2 + 1$ to $(n + 1)^2 - 1$ .
Number of integers between $n^2$ and $(n + 1)^2$ is:

\begin{aligned} \text{Count} &= ((n + 1)^2 - 1) - (n^2 + 1) + 1 \\ &= (n^2 + 2n + 1 - 1) - n^2 \\ &= 2n \end{aligned}

Therefore, there are always $2n$ non-square numbers between $n^2$ and $(n + 1)^2$ .

(i) 16 and 17

Step 1 · Count Numbers Between $16^2$ and $17^2$

Here, $n = 16$ and $n + 1 = 17$ .

Number of numbers between $n^2$ and $(n + 1)^2 = 2n$

\begin{aligned} \text{Number of values} &= 2 \times 16 \\ &= 32 \end{aligned}

Answer

(i) 32

(ii) 99 and 100

Step 1 · Count Numbers Between $99^2$ and $100^2$

Here, $n = 99$ and $n + 1 = 100$ .

Number of numbers between $n^2$ and $(n + 1)^2 = 2n$

\begin{aligned} \text{Number of values} &= 2 \times 99 \\ &= 198 \end{aligned}

Answer

(ii) 198

Common Mistakes

Off-by-One Error: Calculating $(n+1)^2 - n^2 = 2n + 1$ instead of subtracting $1$ more to exclude the boundary $(n+1)^2$ . The count of numbers strictly between is $(n+1)^2 - n^2 - 1 = 2n$ .
Selecting the Larger Number: Multiplying the second number $(n + 1)$ by $2$ instead of the smaller base number $n$ .

More questions in FIO

Which of the following numbers are not perfect squares?

(i) 2032 (ii) 2048 (iii) 1027 (iv) 1089

Which one among $64^2$ , $108^2$ , $292^2$ , $36^2$ has last digit 4?

Given $125^2 = 15625$ , what is the value of $126^2$ ?

(i) $15625 + 126$

(ii) $15625 + 26^2$

(iii) $15625 + 253$

(iv) $15625 + 251$

(v) $15625 + 51^2$

Find the length of the side of a square whose area is $441 \text{ m}^2$ .

Find the smallest square number that is divisible by each of the following numbers: 4, 9, and 10.

Find the smallest number by which 9408 must be multiplied so that the product is a perfect square. Find the square root of the product.

How many numbers lie between the squares of the following numbers?

(i) 16 and 17 (ii) 99 and 100

In the following pattern, fill in the missing numbers:

\begin{aligned} 1^2 + 2^2 + 2^2 &= 3^2 \\ 2^2 + 3^2 + 6^2 &= 7^2 \\ 3^2 + 4^2 + 12^2 &= 13^2 \end{aligned}

(a) $4^2 + 5^2 + 20^2 = (\underline{\quad})^2$

(b) $9^2 + 10^2 + (\underline{\quad})^2 = (\underline{\quad})^2$

How many tiny squares are there in the following picture? Write the prime factorisation of the number of tiny squares.

Q10

Find the cube roots of $27000$ and $10648$ .

Q11

What number will you multiply by 1323 to make it a cube number?

Q12

State true or false. Explain your reasoning.

(i) The cube of any odd number is even.

(ii) There is no perfect cube that ends with 8.

(iii) The cube of a 2-digit number may be a 3-digit number.

(iv) The cube of a 2-digit number may have seven or more digits.

(v) Cube numbers have an odd number of factors.

Q13

You are told that $1331$ is a perfect cube. Can you guess without factorisation what its cube root is? Similarly, guess the cube roots of $4913$ , $12167$ , and $32768$ .

Q14

Which of the following is the greatest? Explain your reasoning.

(i) $67^3 - 66^3$

(ii) $43^3 - 42^3$

(iii) $67^2 - 66^2$

(iv) $43^2 - 42^2$