Finding the Greatest Number with Specific Remainders

The question asks for the greatest number that, when dividing 2675, leaves a remainder of 5, and when dividing 2320, leaves a remainder of 6.

Let the required greatest number be \(N\).

According to the problem statement:

• When 2675 is divided by \(N\), the remainder is 5. This means \(2675 - 5\) is perfectly divisible by \(N\).
• When 2320 is divided by \(N\), the remainder is 6. This means \(2320 - 6\) is perfectly divisible by \(N\).

So, \(N\) must be a common divisor of \(2675 - 5\) and \(2320 - 6\).

Let's calculate these differences:

• \(2675 - 5 = 2670\)
• \(2320 - 6 = 2314\)

The problem asks for the greatest such number. This means \(N\) is the Greatest Common Divisor (GCD) or Highest Common Factor (HCF) of 2670 and 2314.

Calculating the Greatest Common Divisor (GCD)

We can find the GCD of 2670 and 2314 using the Euclidean algorithm.

The steps are as follows:

1. Divide the larger number (2670) by the smaller number (2314) and find the remainder.
2. Replace the larger number with the smaller number, and the smaller number with the remainder.
3. Repeat the process until the remainder is 0. The last non-zero remainder is the GCD.

Let's perform the Euclidean algorithm steps:

• Step 1: Divide 2670 by 2314.

  \(2670 = 1 \times 2314 + 356\)

  The remainder is 356.
• Step 2: Divide 2314 by 356.

  \(2314 = 6 \times 356 + 178\)

  \(6 \times 356 = 2136\), and \(2314 - 2136 = 178\). The remainder is 178.
• Step 3: Divide 356 by 178.

  \(356 = 2 \times 178 + 0\)

  \(2 \times 178 = 356\). The remainder is 0.

Since the remainder is 0 in Step 3, the GCD is the last non-zero remainder, which is 178.

Thus, the GCD of 2670 and 2314 is 178.

Verification

Let's check if dividing 2675 and 2320 by 178 gives the specified remainders:

• \(2675 \div 178\)

  \(2675 = 15 \times 178 + 5\)

  Remainder is 5. Correct.
• \(2320 \div 178\)

  \(2320 = 13 \times 178 + 6\)

  Remainder is 6. Correct.

The number 178 satisfies both conditions, and since it is the GCD of \(2675-5\) and \(2320-6\), it is the greatest such number.

The required greatest number is 178.

Looking at the options, 178 is one of the choices.

Revision Table: Greatest Number & Remainders

Understanding problems involving remainders and greatest common divisors is crucial. Here's a quick summary of the approach for this type of question:

• If a number \(N\) divides \(a\) leaving remainder \(r_1\), then \(N\) divides \(a - r_1\) exactly.
• If a number \(N\) divides \(b\) leaving remainder \(r_2\), then \(N\) divides \(b - r_2\) exactly.
• If you need the greatest such number \(N\), it is the GCD of \((a - r_1)\) and \((b - r_2)\).

Additional Information: Euclidean Algorithm for GCD

The Euclidean Algorithm is an efficient method for computing the greatest common divisor (GCD) of two integers \(a\) and \(b\). The principle is based on the property that the GCD of two numbers does not change if the larger number is replaced by its difference with the smaller number. More formally, for integers \(a\) and \(b\) with \(a > b > 0\), \(\text{GCD}(a, b) = \text{GCD}(b, a \pmod b)\), where \(a \pmod b\) is the remainder when \(a\) is divided by \(b\).

The algorithm continues until the remainder is 0. The GCD is the last non-zero remainder. This method is guaranteed to terminate because the remainders decrease in each step.