Question

Find the greatest number which divides 764 and 1198, leaving remainders 8 and 10 respectively.

Hint:

Using Euclid's division algorithm is a fast way to verify HCF for large integers:
\[ 1188 = 756 \times 1 + 432 \] \[ 756 = 432 \times 1 + 324 \] \[ 432 = 324 \times 1 + 108 \] \[ 324 = 108 \times 3 + 0 \] Since the last non-zero remainder is 108, this confirms that the HCF is indeed 108!

Answer 1

Step 1: Understanding the Question:
The topic of this question is Real Numbers, specifically focusing on the Highest Common Factor (HCF) of integers.
We are looking for the largest positive integer that divides both 764 and 1198 such that the remainder left is 8 and 10 respectively.
This means the required number divides the values \((764 - 8)\) and \((1198 - 10)\) exactly.
Therefore, the required greatest number is the Highest Common Factor (HCF) of these two adjusted values.

Step 2: Key Formula or Approach:
- Find the adjusted numbers:
\[ N_1 = 764 - 8 = 756 \] \[ N_2 = 1198 - 10 = 1188 \] - Calculate the HCF of 756 and 1188 using prime factorization or Euclid's division algorithm.

Step 3: Detailed Explanation:

• Subtract the respective remainders to find the adjusted numbers:
\[ N_1 = 764 - 8 = 756 \] \[ N_2 = 1198 - 10 = 1188 \]

• Find the prime factorization of 756:
\[ 756 = 2 \times 378 \] \[ 756 = 2^2 \times 189 \] \[ 756 = 2^2 \times 3 \times 63 \] \[ 756 = 2^2 \times 3^3 \times 7 = 4 \times 27 \times 7 \]

• Find the prime factorization of 1188:
\[ 1188 = 2 \times 594 \] \[ 1188 = 2^2 \times 297 \] \[ 1188 = 2^2 \times 3 \times 99 \] \[ 1188 = 2^2 \times 3^3 \times 11 = 4 \times 27 \times 11 \]

• Determine the Highest Common Factor (HCF) of 756 and 1188: Identify the lowest power of each common prime factor:
- The lowest power of 2 is \(2^2\).
- The lowest power of 3 is \(3^3\).
Multiply these values to find the HCF:
\[ \text{HCF} = 2^2 \times 3^3 \] \[ \text{HCF} = 4 \times 27 = 108 \]

Step 4: Final Answer:
The greatest number that divides 764 and 1198 leaving remainders 8 and 10 respectively is 108.