1. The problem is to find the greatest length of a wooden scale which can measure 540 cm and 360 cm exactly. 2. This means the scale length must be the greatest common divisor (GCD) of 540 and 360. 3. To find GCD(540, 360), use prime factorization or Euclidean algorithm. 4. Using Euclidean algorithm: $$540 \div 360 = 1 \text{ remainder } 180$$ $$360 \div 180 = 2 \text{ remainder } 0$$ So, GCD is 180 cm. 5. The greatest length of the wooden scale is $\boxed{180}$ cm. 1. Find the greatest number which divides 232 and 305 leaving remainders 7 and 5 respectively. 2. Let the required number be $x$. 3. Since remains are left, subtract the remainders from each number: $$232 - 7 = 225$$ $$305 - 5 = 300$$ 4. The required number $x$ divides 225 and 300 exactly, so find GCD(225, 300). 5. GCD(225, 300): $$300 - 225 = 75$$ GCD(225, 75) = 75 (since 225 \div 75 = 3 remainder 0) 6. So, the greatest number is $\boxed{75}$. 1. Find the largest number which divides 245 and 1029 leaving remainder 5 in each case. 2. Let the number be $x$. 3. Subtract remainder 5: $$245 - 5 = 240$$ $$1029 - 5 = 1024$$ 4. Find GCD(240, 1024). 5. Using Euclidean algorithm: $$1024 \div 240 = 4 \text{ remainder } 64$$ $$240 \div 64 = 3 \text{ remainder } 48$$ $$64 \div 48 = 1 \text{ remainder } 16$$ $$48 \div 16 = 3 \text{ remainder } 0$$ 6. So, GCD = 16. 7. Largest number that divides 245 and 1029 leaving remainder 5 is $\boxed{16}$. 1. Two tankers contain 600 litres and 570 litres of petrol. Find maximum container capacity measuring petrol exactly an integer number of times. 2. This is finding GCD of 600 and 570. 3. Using Euclidean algorithm: $$600 - 570 = 30$$ $$570 \div 30 = 19 \text{ remainder } 0$$ 4. So, GCD = 30 litres. 5. Maximum container capacity is $\boxed{30}$ litres. 1. Length, breadth, height of room are 8 m, 6 m and 4 m. Find longest tape to measure all dimensions exactly. 2. This is the GCD of 8, 6 and 4. 3. GCD(8, 6) = 2 4. GCD(2, 4) = 2 5. Longest tape length is $\boxed{2}$ m.