Find the smallest number which when added by 5 is divisible by 12, 14 and 18.

1. Find the smallest number which when added by 5 is divisible by 12, 14 and 18.

Step 1: Let the smallest number be $N$. According to the problem, $N+5$ must be the Least Common Multiple (LCM) of 12, 14, and 18.

Step 2: Find the prime factorization of each number.

$$12 = 2^2 \times 3$$ $$14 = 2 \times 7$$ $$18 = 2 \times 3^2$$

Step 3: Calculate the LCM by taking the highest power of all prime factors.

$$\text{LCM}(12, 14, 18) = 2^2 \times 3^2 \times 7 = 4 \times 9 \times 7 = 36 \times 7 = 252$$

Step 4: Solve for $N$.

$$N+5 = 252$$ $$N = 252 - 5 = 247$$

The smallest number is $\boxed{\textbf{247}}$.

2. A heap is to be formed with lots of 8, 10 and 15 pebbles of different colours. Find the smallest number of pebbles in the heap.

Step 1: The smallest number of pebbles in the heap must be a common multiple of 8, 10, and 15. Therefore, we need to find the Least Common Multiple (LCM) of these numbers.

Step 2: Find the prime factorization of each number.

$$8 = 2^3$$ $$10 = 2 \times 5$$ $$15 = 3 \times 5$$

Step 3: Calculate the LCM by taking the highest power of all prime factors.

$$\text{LCM}(8, 10, 15) = 2^3 \times 3 \times 5 = 8 \times 3 \times 5 = 120$$

The smallest number of pebbles in the heap is $\boxed{\textbf{120}}$.

3. Three measuring tapes are 64 cm, 72 cm and 96 cm. What is the shortest length that can be measured by any one of the tapes exactly?

Step 1: The shortest length that can be measured exactly by any of the tapes must be a common multiple of their lengths. We need to find the Least Common Multiple (LCM) of 64, 72, and 96.

Step 2: Find the prime factorization of each number.

$$64 = 2^6$$ $$72 = 2^3 \times 3^2$$ $$96 = 2^5 \times 3$$

Step 3: Calculate the LCM by taking the highest power of all prime factors.

$$\text{LCM}(64, 72, 96) = 2^6 \times 3^2 = 64 \times 9 = 576$$

The shortest length is $\boxed{\textbf{576 cm}}$.

4. Find the greatest number that will exactly divide 73 and 117.

Step 1: We need to find the Greatest Common Divisor (GCD) or Highest Common Factor (HCF) of 73 and 117.

Step 2: Find the prime factorization of each number.

$$73 = 73^1 \quad (\text{73 is a prime number})$$ $$117 = 3 \times 39 = 3 \times 3 \times 13 = 3^2 \times 13$$

Step 3: Identify common prime factors.
There are no common prime factors between 73 and 117.

Step 4: When there are no common prime factors, the HCF is 1.

$$\text{HCF}(73, 117) = 1$$

The greatest number is $\boxed{\textbf{1}}$.

5. Two ropes of length 28 m and 36 m are to be cut into bits of same length. Find the greatest possible length for each bit.

Step 1: The greatest possible length for each bit must be a common divisor of both 28 m and 36 m. We need to find the Greatest Common Divisor (GCD) or Highest Common Factor (HCF) of 28 and 36.

Step 2: Find the prime factorization of each number.

$$28 = 2^2 \times 7$$ $$36 = 2^2 \times 3^2$$

Step 3: Calculate the HCF by taking the lowest power of common prime factors.

$$\text{HCF}(28, 36) = 2^2 = 4$$

The greatest possible length for each bit is $\boxed{\textbf{4 m}}$.

6. Find the largest size of handkerchiefs which can be cut from a cloth having length 2.75 m and breadth 75 cm.

Step 1: Convert all measurements to the same unit. Convert meters to centimeters.

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