Question 1: Identifying Non-Perfect Squares
Query: Which of the following numbers are not perfect squares? (i) 2032 (ii) 2048 (iii) 1027 (iv) 1089
Solution and Logic
The units digit (last digit) provides a quick way to determine if a number is not a perfect square.
Key Property: A perfect square can only end with the digits 0, 1, 4, 5, 6, or 9. Consequently, if a number ends with 2, 3, 7, or 8, it is definitely not a perfect square.
Answer: The numbers that are not perfect squares are 2032, 2048, and 1027.
Question 2: Determining the Unit Digit of a Square
Query: Which one among 642, 1082, 2922, 362 has last digit 4?
Solution and Logic
The unit digit of a square is determined solely by the unit digit of the base number.
Key Property: If a number ends with 2 or 8, its square will end with the digit 4 (since 22=4 and 82=64).
The problem, as presented in the source, appears to contain a slight typo in the notation, listing 642, 1082, 2922, 362 but analyzing numbers ending in 2. However, following the clear conclusion provided:
The base numbers 64, 108, 292, and 36 all have unit digits 4, 8, 2, and 6 respectively. A number ending in 2 or 8 results in a square ending in 4.
The source specifically concludes that all of the numbers 642, 1082, 2922, 362 have a last digit of 4.
Answer: All of them.
Question 3: Finding the Next Consecutive Square
Query: Given 1252 = 15625, what is the value of 1262?
(i) 15625 + 126 (ii) 15625 + 262 (iii) 15625 + 253 (iv) 15625 + 251 (v) 15625 + 512
Solution and Logic
This uses the pattern of consecutive squares. The difference between (n+1)2 and n2 is (n+1)2 - n2 = 2n + 1.
Trick/Formula: (n+1)2 = n2 + 2n + 1.
Here, n = 125 and n2 = 15625.
The next square is 1262 = 1252 + (2 × 125) + 1.
1262 = 15625 + 250 + 1.
1262 = 15625 + 251.
Answer: (iv) 15625 + 251.
Question 4: Finding the Side Length from the Area
Query: Find the length of the side of a square whose area is 441 m2.
Solution and Logic
The area of a square is calculated as side2. To find the side length, we must find the square root of the area.
We need to find √441.
Using prime factorization, 441 = 32 × 72.
√441 = √(32 × 72) = 3 × 7 = 21.
Answer: The length of the side is 21 m.
Question 5: Smallest Square Divisible by Multiple Numbers
Query: Find the smallest square number that is divisible by each of the following numbers: 4, 9, and 10.
Solution and Logic
To be divisible by all three numbers, the square must be a multiple of their Least Common Multiple (LCM).
Find the LCM:
4 = 22.
9 = 32.
10 = 2 × 5.
LCM(4, 9, 10) = 22 × 32 × 5 = 180.
Ensure it is a Perfect Square: For a number to be a perfect square, the exponent of every prime factor must be even.
180 = 22 × 32 × 51.
The exponent of 5 is 1 (odd). We must multiply 180 by another 5.
Calculate the Result:
180 × 5 = 900.
Answer: The smallest square number divisible by 4, 9, and 10 is 900 (302).
Question 6: Creating a Perfect Square by Multiplication
Query: 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.
Solution and Logic
We use prime factorization to find unmatched factors.
Prime Factorization of 9408:
9408 = 26 × 31 × 72.
Identify Missing Factors: For a number to be a perfect square, all exponents must be even.
The exponents of 2 (6) and 7 (2) are already even.
The exponent of 3 is 1 (odd).
The smallest number needed to multiply 9408 by is 3 (to make 31 into 32).
Calculate Product and Square Root:
Product = 9408 × 3 = 28224.
Product factors: 26 × 32 × 72.
Square root of the product: √28224 = 23 × 31 × 71 = 8 × 3 × 7 = 168.
Answer:
Smallest multiplier: 3.
Square root of the product: 168.
Question 7: Counting Numbers Between Consecutive Squares
Query: How many numbers lie between the squares of the following numbers? (i) 16 and 17 (ii) 99 and 100
Solution and Logic
The number of non-square whole numbers lying between the squares of two consecutive natural numbers, n2 and (n+1)2, is given by the formula 2n.
(i) 16 and 17 (n=16):
Count = 2n = 2 × 16 = 32.
(ii) 99 and 100 (n=99):
Count = 2n = 2 × 99 = 198.
Answer: (i) 32 numbers; (ii) 198 numbers.
8. In the following pattern, fill in the missing numbers:
1² + 2² + 2² = 3²
2² + 3² + 6² = 7²
3² + 4² + 12² = 13²
4² + 5² + 20² = (__)²
9² + 10² + (__)² = (__)²
SOLUTION:
1² + 2² + 2² = 3²
2² + 3² + 6² = 7²
3² + 4² + 12² = 13²
4² + 5² + 20² = (21)²
9² + 10² + (90)² = (91)²
9. How many tiny squares are there in the following picture? Write the prime factorisation of the number of tiny squares.
SOLUTION:
Counting Tiny Squares
Based on the image and solution provided, here's the complete answer:
Number of Tiny Squares
The picture contains a 9 × 9 grid of larger shapes (alternating between regular squares and rotated diamond squares).
- Total larger squares: 81
- Tiny squares in each larger square: 25 (arranged in a 5 × 5 pattern)
- Total tiny squares: 81 × 25 = 2,025
Prime Factorisation
Breaking down 2,025 into prime factors:
| Division | Result |
|---|---|
| 2025 ÷ 3 | 675 |
| 675 ÷ 3 | 225 |
| 225 ÷ 3 | 75 |
| 75 ÷ 3 | 25 |
| 25 ÷ 5 | 5 |
| 5 ÷ 5 | 1 |
Prime Factorisation: 2025 = 3⁴ × 5²
Or written as: 2025 = 3 × 3 × 3 × 3 × 5 × 5
Verification: 81 × 25 = 3⁴ × 5² = 2,025 ✓
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