1. Problem: Find the greatest length of a wooden scale which can be used to measure 540 cm and 360 cm exactly. Step 1: We need to find the Greatest Common Divisor (GCD) of 540 and 360. Step 2: Prime factorize each number: - 540 = $2^2 \times 3^3 \times 5$ - 360 = $2^3 \times 3^2 \times 5$ Step 3: Take the minimum power of each prime common to both: - For 2: minimum of 2 and 3 is 2 - For 3: minimum of 3 and 2 is 2 - For 5: minimum of 1 and 1 is 1 Step 4: Calculate GCD: $$\text{GCD} = 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180$$ Answer: The greatest length is 180 cm. 2. Problem: Find the greatest number which divides 232 and 305 leaving 7 and 5 as remainders respectively. Step 1: Let the number be $x$. Step 2: Then 232 leaves remainder 7 means $x$ divides $(232 - 7) = 225$ exactly. Similarly, 305 leaves remainder 5 means $x$ divides $(305 - 5) = 300$ exactly. Step 3: Find GCD of 225 and 300. Prime factorize: - 225 = $3^2 \times 5^2$ - 300 = $2^2 \times 3 \times 5^2$ Minimum powers: - For 2: 0 (not common) - For 3: minimum of 2 and 1 is 1 - For 5: minimum of 2 and 2 is 2 GCD: $$3^1 \times 5^2 = 3 \times 25 = 75$$ Answer: The required greatest number is 75. 3. Problem: Find the largest number which divides 245 and 1029 leaving remainder 5 in each case. Step 1: Let that number be $x$. Step 2: Then $x$ divides $245 - 5 = 240$ and $1029 - 5 = 1024$ exactly. Step 3: Find GCD of 240 and 1024. Prime factorize: - 240 = $2^4 \times 3 \times 5$ - 1024 = $2^{10}$ Common minimum powers: - For 2: minimum of 4 and 10 is 4 - For 3: 0 (not common) - For 5: 0 (not common) So GCD is: $$2^4 = 16$$ Answer: The largest number is 16. 4. Problem: Two tankers contain 600 litres and 570 litres of petrol respectively. Find the maximum capacity of the container which can measure the petrol of either tanker in exact number of times. Step 1: Find the GCD of 600 and 570. Prime factorize: - 600 = $2^3 \times 3 \times 5^2$ - 570 = $2 \times 3 \times 5 \times 19$ Minimum power: - For 2: minimum of 3 and 1 is 1 - For 3: minimum of 1 and 1 is 1 - For 5: minimum of 2 and 1 is 1 - For 19: 0 (not common) GCD: $$2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30$$ Answer: Maximum container capacity is 30 litres. 5. Problem: The length, breadth and height of a room are 8 m, 6 m and 4 m respectively. Determine the longest tape which can measure all these dimensions exactly. Step 1: Find GCD of 8, 6, and 4. Prime factorize: - 8 = $2^3$ - 6 = $2 \times 3$ - 4 = $2^2$ Minimum powers: - For 2: minimum of 3, 1, and 2 is 1 - For 3: 0 (not common to all) GCD: $$2^1 = 2$$ Answer: The longest tape length is 2 meters.