A Square and A Cube MATHS - A Square and A Cube

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?

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

Consider the following numbers and their squares.

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

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

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

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

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

Does every number have an even number of factors?

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

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

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.

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.

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

Which of the following numbers have the digit 6 in the units place?
82²

Which of the following numbers have the digit 6 in the units place?
74²

Which of the following numbers have the digit 6 in the units place?
56²

Which of the following numbers have the digit 6 in the units place?
46²

Find the squares of the : 30² = __________

Find the squares of the : 29² = __________

Find the squares of the : 28² = __________

Find the squares of the : 27² = __________

Find the squares of the : 26² = __________

Find the cube roots of 27000 and 10648.

Which of the following numbers are not perfect squares?
(i) 2032
(ii) 2048
(iii) 1027
(iv) 1089

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?