Greatest Number Problem Explained

The question asks for the greatest number, denoted as N, that divides 600, 631, and 724, ensuring the remainder is identical for each division.

Number N Differences Method

A key principle states that if a number N divides multiple numbers (say A, B, C) leaving the same remainder, then N must perfectly divide the differences between these numbers (A-B, B-C, A-C). We first calculate these differences:

• Difference between 631 and 600: \(631 - 600 = 31\)
• Difference between 724 and 631: \(724 - 631 = 93\)
• Difference between 724 and 600: \(724 - 600 = 124\)

HCF Calculation for N

The greatest number N we are looking for is the Highest Common Factor (HCF) of these calculated differences: \(31\), \(93\), and \(124\).

To find the HCF, we list the prime factors for each difference:

• Factors of \(31\): \(31\) (as it's a prime number)
• Factors of \(93\): \(3 \times 31\)
• Factors of \(124\): \(2 \times 2 \times 31\), which simplifies to \(4 \times 31\)

By comparing the factors, we see that \(31\) is the only common factor present in all three numbers. Therefore, the HCF(\(31\), \(93\), \(124\)) is \(31\).

Value of N Determined

The HCF represents the greatest number N that satisfies the condition.

Thus, the value of N is \(31\).