Finding the Greatest Divisor with Remainders

The problem asks for the largest number that divides two numbers, 209 and 347, leaving specific remainders. This means the number must divide evenly into the original numbers after their respective remainders are subtracted.

Step 1: Subtract Remainders

First, subtract the remainder from each number:

• For 209 with remainder 5: $209 - 5 = 204$
• For 347 with remainder 7: $347 - 7 = 340$

The number we are looking for must be a common divisor of 204 and 340.

Step 2: Calculate the Highest Common Factor (HCF)

To find the greatest number, we need to calculate the Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of 204 and 340.

We can use prime factorization:

• Prime factors of 204: $2 \times 102 = 2 \times 2 \times 51 = 2 \times 2 \times 3 \times 17 = 2^2 \times 3 \times 17$
• Prime factors of 340: $10 \times 34 = (2 \times 5) \times (2 \times 17) = 2 \times 2 \times 5 \times 17 = 2^2 \times 5 \times 17$

The common prime factors are $2^2$ and $17$. The HCF is the product of these common factors:

HCF(204, 340) = $2^2 \times 17 = 4 \times 17 = 68$

Conclusion

The greatest number that will divide 209 leaving a remainder of 5 and 347 leaving a remainder of 7 is 68.