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A Square and a Cube

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Page 1: • Person 1 opens every locker. • Person 2 toggles every 2nd locker (i.e., closes it if it is open, opens it if it is…
Page 2: Does every number have an even number of factors?
Page 2: Can you use this insight to find more numbers with an odd number of factors? Hence all the following numbers have an…
Page 3: Write the locker numbers that remain open.
Page 3: Which are these five lockers?
Page 3: Why are the numbers, 1, 4, 9, 16, …, called squares? We know that the number of unit squares in a square (the area…
Page 4: The squares of natural numbers are called perfect squares. For example, 1, 4, 9, 16, 25, … are all perfect squares.…
Page 4: Study the squares in the table above. What are the digits in the units places of these numbers? All these numbers…
Page 5: Which of the following numbers have the digit 6 in the units place?
Page 5: Find more such patterns by observing the numbers and their squares from the table you filled earlier. Consider the…
Page 5: What can you say about the parity of a number and its square?
Page 5: Let us explore the differences between consecutive squares. What do you notice? See if this pattern continues for…
Page 6: Using the pattern above, find 36², given that 35²=1225. How do we find the 36th odd number? What is the nᵗʰ odd number?
Page 6: Consider a number such as 38 that is not a square and subtract consecutive odd numbers starting from 1. This shows…
Page 7: Find how many numbers lie between two consecutive perfect squares. Do you notice a pattern?
Page 7: How many square numbers are there between 1 and 100? How many are between 101 and 200? Using the table of squares…
Page 7: Do you remember triangular numbers? Can you see any relation between triangular numbers and square numbers? Extend…
Page 7: The area of a square is 49 sq. cm. What is the length of its side?
Page 8: What is the square root of 64?
Page 8: Given a number, such as 576 or 327, how do we find out if it is a perfect square? If it is a perfect square, how can…
Page 9: 2. Recall that every square can be expressed as a sum of consecutive odd numbers starting from 1. Consider √(81).…
Page 9: 3. We know that a perfect square is obtained by multiplying an integer by itself. Will looking at a number’s prime…
Page 9: Find whether 1156 and 2800 are perfect squares using prime factorisation.
Page 9: We can estimate the square root of larger perfect squares by looking at the closest perfect squares we are familiar…
Page 10: Consider the following situations — Aribam and Bijou play a game. One says a number and the other replies with its…
Page 10: Here is another problem that requires estimating square roots. Akhil has a square piece of cloth of area 125 cm². He…
Page 10: Which of the following numbers are not perfect squares?
Page 10: Which one among 64², 108², 292², 36² has last digit 4?
Page 10: Given 125²=15625, what is the value of 126²?
Page 10: Find the length of the side of a square whose area is 441 m².
Page 10: Find the smallest square number that is divisible by each of the following numbers: 4, 9, and 10.
Page 10: Find the smallest number by which 9408 must be multiplied so that the product is a perfect square. Find the square…
Page 10: How many numbers lie between the squares of the following numbers?
Page 10: In the following pattern, fill in the missing numbers:
Page 11: 9. How many tiny squares are there in the following picture? Write the prime factorisation of the number of tiny…
Page 11: You know the word cube from geometry. A cube is a solid figure all of whose all sides meet at right angles and are…
Page 12: Is 9 a cube?
Page 12: Can you estimate the number of unit cubes in a cube with an edge length of 4 units?
Page 12: 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…
Page 13: Similar to squares, can you find the number of cubes with 1 digit, 2 digits, and 3 digits? What do you observe?
Page 13: Can a cube end with exactly two zeroes (00)? Explain.
Page 13: The next two taxicab numbers after 1729 are 4104 and 13832. Find the two ways in which each of these can be…
Page 14: Consecutive odd numbers have a role to play with cubes too. Look at the following pattern: Later in this series, we…
Page 14: How do we find out if a number is a cube? Taking inspiration from the case of squares, let us see if we can use…
Page 15: Find the cube roots of these numbers:
Page 15: Compute successive differences over levels for perfect cubes until all the differences at a level are the same. What…
Page 15: The first known list of perfect squares and perfect cubes was compiled by the Babylonians as far back as 1700 BCE.…
Page 16: 1. Find the cube roots of 27000 and 10648.
Page 16: 2. What number will you multiply by 1323 to make it a cube number?
Page 16: 3. State true or false. Explain your reasoning. (i) The cube of any odd number is even. (ii) There is no perfect…
Page 16: 4. You are told that 1331 is a perfect cube. Can you guess without factorisation what its cube root is? Similarly,…
Page 17: 5. Which of the following is the greatest? Explain your reasoning.
Page 18: Look at the following numbers: 3 6 10 15 1 They are arranged such that each pair of adjacent numbers adds up to a…

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