Figure it out-1

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1. Which of the following numbers are not perfect squares?

Property: Perfect squares can only end with the digits 0, 1, 4, 5, 6, or 9 at their unit place. Any number ending in 2, 3, 7, or 8 is definitely not a perfect square.

• (i) 2032: Ends in 2. Therefore, it is not a perfect square.
• (ii) 2048: Ends in 8. Therefore, it is not a perfect square.
• (iii) 1027: Ends in 7. Therefore, it is not a perfect square.
• (iv) 1089: Ends in 9. Since 33 × 33 = 1089, it is a perfect square.

Answer: (i) 2032, (ii) 2048, and (iii) 1027 are not perfect squares.

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2. Which one among 64², 108², 292², 36² has last digit 4?

To find the last digit of a square number, we only need to square its unit digit:

• For 64²: Last digit of 4² = 16 is 6
• For 108²: Last digit of 8² = 64 is 4
• For 292²: Last digit of 2² = 4 is 4
• For 36²: Last digit of 6² = 36 is 6

Answer: Both 108² and 292² have 4 as their last digit.

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3. Given 125² = 15625, what is the value of 126²?

Property: The difference between the squares of two consecutive numbers n and (n+1) is given by: (n+1)² - n² = (n+1) + n = 2n + 1.

Therefore, (n+1)² = n² + (n+1) + n

Let n = 125:

126² = 125² + 126 + 125

126² = 15625 + 251

Answer: The correct option is (iv) 15625 + 251.

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4. Find the length of the side of a square whose area is 441 m².

Formula: Area of a square = (side)²

Given Area = 441 m²

side = √441

By prime factorization: 441 = 3 × 3 × 7 × 7 = 3² × 7²

side = 3 × 7 = 21 m

Answer: The length of the side of the square is 21 m.

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

Step 1: Find the Least Common Multiple (LCM) of 4, 9, and 10.

• Prime factorization of 4 = 2 × 2
• Prime factorization of 9 = 3 × 3
• Prime factorization of 10 = 2 × 5

LCM(4, 9, 10) = 2 × 2 × 3 × 3 × 5 = 180

Step 2: Make the LCM a perfect square by pairing the factors.

In the prime factorization of 180 = (2 × 2) × (3 × 3) × 5, the prime factor 5 does not have a pair.

To make it a perfect square, we must multiply 180 by 5:

Required Square Number = 180 × 5 = 900

Answer: The smallest square number is 900.

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6. 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.

Step 1: Find the prime factorization of 9408.

9408 = 2 × 2 × 2 × 2 × 2 × 2 × 3 × 7 × 7

Grouping into pairs: 9408 = (2 × 2) × (2 × 2) × (2 × 2) × 3 × (7 × 7)

The prime factor 3 is left unpaired.

Therefore, the smallest number to be multiplied is 3.

Step 2: Find the new product and its square root.

New Product = 9408 × 3 = 28224

√28224 = 2 × 2 × 2 × 3 × 7 = 168

Answer: The smallest number to multiply by is 3, and the square root of the product is 168.

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

Property: There are 2n non-perfect square numbers lying between the squares of consecutive numbers n and (n+1).

(i) 16 and 17

Here, n = 16

Number of elements = 2n = 2 × 16 = 32

Answer: There are 32 numbers lying between 16² and 17².

(ii) 99 and 100

Here, n = 99

Number of elements = 2n = 2 × 99 = 198

Answer: There are 198 numbers lying between 99² and 100².

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8. In the following pattern, fill in the missing numbers:

Analysis of the Pattern:

Let the general form of each row be: a² + b² + c² = d²

• The third number 'c' is always the product of the first two numbers 'a' and 'b'. That is: c = a × b
• The fourth number 'd' on the right-hand side is always 1 more than the third number 'c'. That is: d = c + 1

Let's verify with the given complete rows:

• Row 1: 1² + 2² + 2² = 3² → Here, 1 × 2 = 2 (third number) and 2 + 1 = 3 (fourth number).
• Row 2: 2² + 3² + 6² = 7² → Here, 2 × 3 = 6 (third number) and 6 + 1 = 7 (fourth number).
• Row 3: 3² + 4² + 12² = 13² → Here, 3 × 4 = 12 (third number) and 12 + 1 = 13 (fourth number).

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Finding the Missing Numbers:

Row 4: 4² + 5² + 20² = (___)²

Following the rule d = c + 1, where c = 20:

d = 20 + 1 = 21

Completed Row: 4² + 5² + 20² = 21²

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Row 5: 9² + 10² + (___)² = (___)²

Following the rule c = a × b, where a = 9 and b = 10:

c = 9 × 10 = 90

Following the rule d = c + 1, where c = 90:

d = 90 + 1 = 91

Completed Row: 9² + 10² + 90² = 91²

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

Step 1: Count the total number of large blocks in the grid.

• Number of horizontal rows of blocks = 9
• Number of vertical columns of blocks = 9
• Total number of blocks = 9 × 9 = 81 blocks

Step 2: Count the number of tiny squares inside each block.

• Each individual block (whether straight or tilted) contains a 4 × 4 sub-grid.
• Number of tiny squares per block = 4 × 4 = 16 tiny squares

Step 3: Calculate the total number of tiny squares.

Total tiny squares = Total blocks × Tiny squares per block

Total tiny squares = 81 × 16 = 1296

Step 4: Find the prime factorization of 1296.

We divide 1296 continuously by prime numbers:

• 1296 ÷ 2 = 648
• 648 ÷ 2 = 324
• 324 ÷ 2 = 162
• 162 ÷ 2 = 81
• 81 ÷ 3 = 27
• 27 ÷ 3 = 9
• 9 ÷ 3 = 3
• 3 ÷ 3 = 1

Answer:

• The total number of tiny squares is 1296.
• The prime factorization of 1296 is: 2 × 2 × 2 × 2 × 3 × 3 × 3 × 3 = 2⁴ × 3⁴

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Figure it out-2

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1. Find the cube roots of 27000 and 10648.

For 27000:

By prime factorization:

27000 = 2 × 2 × 2 × 3 × 3 × 3 × 5 × 5 × 5

Grouping the factors in triplets: 27000 = (2³) × (3³) × (5³)

Taking one factor from each triplet: Cube root of 27000 = 2 × 3 × 5 = 30

For 10648:

By prime factorization:

10648 = 2 × 2 × 2 × 11 × 11 × 11

Grouping the factors in triplets: 10648 = (2³) × (11³)

Taking one factor from each triplet: Cube root of 10648 = 2 × 11 = 22

Answer: Cube root of 27000 is 30 and cube root of 10648 is 22.

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

Step 1: Find the prime factorization of 1323.

1323 = 3 × 3 × 3 × 7 × 7 = (3 × 3 × 3) × (7 × 7)

Step 2: Observe the triplets.

The prime factor 3 forms a complete triplet, but the prime factor 7 appears only twice.

To make it a perfect cube, we need one more 7 to complete the triplet.

Answer: The smallest number to multiply by is 7.

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3. State true or false. Explain your reasoning.

• (i) The cube of any odd number is even.
  False. The cube of an odd number is always odd. For example, 3³ = 27 (which is odd).
• (ii) There is no perfect cube that ends with 8.
  False. Perfect cubes can end with 8. For example, 2³ = 8 and 12³ = 1728.
• (iii) The cube of a 2-digit number may be a 3-digit number.
  False. The smallest 2-digit number is 10, and its cube is 10³ = 1000, which has 4 digits.
• (iv) The cube of a 2-digit number may have seven or more digits.
  False. The largest 2-digit number is 99, and its cube is 99³ = 970299, which has 6 digits.
• (v) Cube numbers have an odd number of factors.
  False. Perfect squares have an odd number of factors. Perfect cubes can have an even number of factors. For example, 8 has four factors: 1, 2, 4, 8.

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4. 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.

Estimation Method Rules: Group the number into two parts starting from the rightmost digit. The first group gives the unit digit of the cube root, and the second group gives the tens digit.

• For 1331: Groups are (1) and (331). The group (331) ends in 1, so the unit digit is 1. The other group is 1, and 1³ = 1, so the tens digit is 1.
  Cube root = 11
• For 4913: Groups are (4) and (913). The group (913) ends in 3, so the unit digit is 7 (since 7³ = 343). The other group is 4, which lies between 1³ = 1 and 2³ = 8. Taking the smaller value, the tens digit is 1.
  Cube root = 17
• For 12167: Groups are (12) and (167). The group (167) ends in 7, so the unit digit is 3 (since 3³ = 27). The other group is 12, which lies between 2³ = 8 and 3³ = 27. Taking the smaller value, the tens digit is 2.
  Cube root = 23
• For 32768: Groups are (32) and (768). The group (768) ends in 8, so the unit digit is 2 (since 2³ = 8). The other group is 32, which lies between 3³ = 27 and 4³ = 64. Taking the smaller value, the tens digit is 3.
  Cube root = 32

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

Let's calculate the values for each expression:

• (i) 67³ - 66³:
  Using the identity x³ - y³ = (x - y)(x² + xy + y²):
  67³ - 66³ = (67 - 66) × (67² + 67 × 66 + 66²) = 1 × (4489 + 4422 + 4356) = 13267
• (ii) 43³ - 42³:
  43³ - 42³ = (43 - 42) × (43² + 43 × 42 + 42²) = 1 × (1849 + 1806 + 1764) = 5419
• (iii) 67² - 66²:
  Using x² - y² = (x - y)(x + y):
  67² - 66² = (67 - 66) × (67 + 66) = 1 × 133 = 133
• (iv) 43² - 42²:
  43² - 42² = (43 - 42) × (43 + 42) = 1 × 85 = 85

Reasoning: Comparing all the calculated values (13267, 5419, 133, 85), 13267 is clearly the largest value because the cubes of larger consecutive numbers grow much faster than smaller bases or squares.

Answer: (i) 67³ - 66³ is the greatest.