Rational Numbers

Context: Look at the following numbers: 3 6 10 15 1. They are arranged such that each pair of adjacent numbers adds up to a square: 3 + 6 = 9, 6 + 10 = 16, 10 + 15 = 25, 15 + 1 = 16.

Q. Try arranging the numbers 1 to 17 (without repetition) in a row in a similar way — the sum of every adjacent pair of numbers should be a square. Can you arrange them in more than one way? If not, can you explain why?

Context: Look at the following numbers: 3 6 10 15 1. They are arranged such that each pair of adjacent numbers adds up to a square: 3 + 6 = 9, 6 + 10 = 16, 10 + 15 = 25, 15 + 1 = 16.

Q. Can you do the same with numbers from 1 to 32 (again, without repetition), but this time arranging all the numbers in a circle?

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:

$1^2 + 2^2 + 2^2 = 3^2$ $2^2 + 3^2 + 6^2 = 7^2$ $3^2 + 4^2 + 12^2 = 13^2$

(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.

Find the cube roots of 27000 and 10648.

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

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.

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.

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) $43^2 - 42^2$

Context: Queen Ratnamanjuri left a puzzle in her will for her son Khoisnam and 99 relatives. They are in a room with 100 lockers, numbered 1 to 100.

• Person 1 opens every locker.
• Person 2 toggles every 2nd locker (closes if open, opens if closed).
• Person 3 toggles every 3rd locker (3rd, 6th, 9th, ...).
• Person 4 toggles every 4th locker (4th, 8th, 12th, ...). This continues until all 100 get their turn.

Q. Before the process begins, Khoisnam realises that he already knows which lockers will be open at the end. How did he figure out the answer?

Hint: Find out how many times each locker is toggled.

Does every number have an even number of factors?

Can you use this insight to find more numbers with an odd number of factors?

Context: In a room with 100 lockers, only lockers whose numbers are square numbers remain open.

Q. Write the locker numbers that remain open.

Find the squares of the first 30 natural numbers and fill in the table below.

Context: Patterns and Properties of Perfect Squares

Find the squares of the first 30 natural numbers and fill in the table below.

Q. What patterns do you notice? Share your observations and make conjectures.

If a number ends in 0, 1, 4, 5, 6 or 9, is it always a square?

Write 5 numbers such that you can determine by looking at their units digit that they are not squares.

Let us consider square numbers ending in 6: $16 = 4^2$, $36 = 6^2$, $196 = 14^2$, $256 = 16^2$, $576 = 24^2$, and $676 = 26^2$. Which of the following numbers have the digit 6 in the units place?

(i) $38^2$ (ii) $34^2$ (iii) $46^2$ (iv) $56^2$ (v) $74^2$ (vi) $82^2$

Find more such patterns by observing the numbers and their squares from the table you filled earlier.

If a number contains 3 zeros at the end, how many zeros will its square have at the end?

What do you notice about the number of zeros at the end of a number and the number of zeros at the end of its square? Will this always happen? Can we say that squares can only have an even number of zeros at the end?

What can you say about the parity of a number and its square?

Find how many numbers lie between two consecutive perfect squares. Do you notice a pattern?

How many square numbers are there between 1 and 100? How many are between 101 and 200? Using the table of squares you filled earlier, enter the values below, tabulating the number of squares in each block of 100. What is the largest square less than 1000?

Can you see any relation between triangular numbers and square numbers? Extend the pattern shown and draw the next term.

Find whether 1156 and 2800 are perfect squares using prime factorisation.

How many cubes of side 1 cm make a cube of side 2 cm?

How many cubes of side 1 cm will make a cube of side 3 cm?

Is 9 a cube?

Can you estimate the number of unit cubes in a cube with an edge length of 4 units?

Complete the table below.

What patterns do you notice in the table above?

We know that 0, 1, 4, 5, 6, 9 are the only last digits possible for squares. What are the possible last digits of cubes?

Similar to squares, can you find the number of cubes with 1 digit, 2 digits, and 3 digits? What do you observe?

Can a cube end with exactly two zeroes (00)? Explain.

The next two taxicab numbers after 1729 are 4104 and 13832. Find the two ways in which each of these can be expressed as the sum of two positive cubes.

Context: Look at the following pattern of consecutive odd numbers: $1 = 1 = 1^3$ $3 + 5 = 8 = 2^3$ $7 + 9 + 11 = 27 = 3^3$ $13 + 15 + 17 + 19 = 64 = 4^3$ $21 + 23 + 25 + 27 + 29 = 125 = 5^3$ $31 + 33 + 35 + 37 + 39 + 41 = 216 = 6^3$

Later in this series, we get the following set of consecutive numbers: $91 + 93 + 95 + 97 + 99 + 101 + 103 + 105 + 107 + 109$

Q. Can you tell what this sum is without doing the calculation?

Find the cube roots of these numbers:

(i) $\sqrt[3]{64} =$ (ii) $\sqrt[3]{512} =$ (iii) $\sqrt[3]{729} =$

Compute successive differences over levels for perfect cubes until all the differences at a level are the same. What do you notice?