Finding the Greatest Number that Exactly Divides Multiple Numbers

The question asks us to find the greatest number that exactly divides 2880, 6525, and 8307. This number is also known as the Greatest Common Divisor (GCD) or Highest Common Factor (HCF) of these three numbers.

To find the greatest number that exactly divides a set of numbers, we look for the largest number that is a common factor for all of them. We can approach this by checking the given options or by using methods like prime factorization or the Euclidean algorithm. Since we have options, testing them is a straightforward method.

Checking Options for Divisibility

Let's test each option to see if it exactly divides 2880, 6525, and 8307.

Testing Option 1: 7

• Divide 2880 by 7: \( \frac{2880}{7} \approx 411.43 \)

Since 2880 is not exactly divisible by 7, 7 is not the greatest number that exactly divides all three numbers.

Testing Option 2: 11

• Divide 2880 by 11: \( \frac{2880}{11} \approx 261.82 \)

Since 2880 is not exactly divisible by 11, 11 is not the greatest number that exactly divides all three numbers.

Testing Option 3: 3

We can use the divisibility rule for 3. A number is divisible by 3 if the sum of its digits is divisible by 3.

• For 2880: Sum of digits = \( 2 + 8 + 8 + 0 = 18 \). 18 is divisible by 3, so 2880 is divisible by 3.
• For 6525: Sum of digits = \( 6 + 5 + 2 + 5 = 18 \). 18 is divisible by 3, so 6525 is divisible by 3.
• For 8307: Sum of digits = \( 8 + 3 + 0 + 7 = 18 \). 18 is divisible by 3, so 8307 is divisible by 3.

Since 3 exactly divides all three numbers, it is a common divisor.

Testing Option 4: 9

We can use the divisibility rule for 9. A number is divisible by 9 if the sum of its digits is divisible by 9.

• For 2880: Sum of digits = \( 2 + 8 + 8 + 0 = 18 \). 18 is divisible by 9, so 2880 is divisible by 9. \( \frac{2880}{9} = 320 \)
• For 6525: Sum of digits = \( 6 + 5 + 2 + 5 = 18 \). 18 is divisible by 9, so 6525 is divisible by 9. \( \frac{6525}{9} = 725 \)
• For 8307: Sum of digits = \( 8 + 3 + 0 + 7 = 18 \). 18 is divisible by 9, so 8307 is divisible by 9. \( \frac{8307}{9} = 923 \)

Since 9 exactly divides all three numbers, it is a common divisor. Comparing the common divisors found so far (3 and 9), 9 is greater than 3.

Based on the options provided, 9 is the greatest number that exactly divides 2880, 6525, and 8307.

Summary of Divisibility Checks

The table shows that only 3 and 9 are common divisors among the options. The greatest among these is 9.

Revision Table - Greatest Common Divisor

Additional Information - Finding GCD

While checking options worked well here, for numbers where options aren't given or are too numerous, other methods are used to find the greatest number that exactly divides the numbers:

• Prime Factorization Method: Find the prime factorization of each number. The GCD is the product of the common prime factors, each raised to the lowest power it appears in any of the factorizations.
  • Example for two numbers: Find GCD of 12 and 18. \( 12 = 2^2 \times 3^1 \), \( 18 = 2^1 \times 3^2 \). Common factors are 2 and 3. Lowest power of 2 is \( 2^1 \). Lowest power of 3 is \( 3^1 \). GCD = \( 2^1 \times 3^1 = 6 \).
• Euclidean Algorithm: This method is efficient for finding the GCD of two numbers. To find the GCD of three numbers (a, b, c), first find GCD(a, b). Then find GCD(GCD(a, b), c).
  • Example for two numbers: Find GCD of 48 and 18.
    1. Divide 48 by 18: \( 48 = 2 \times 18 + 12 \) (Remainder is 12)
    2. Replace the larger number with the smaller number, and the smaller number with the remainder: Find GCD of 18 and 12.
    3. Divide 18 by 12: \( 18 = 1 \times 12 + 6 \) (Remainder is 6)
    4. Replace: Find GCD of 12 and 6.
    5. Divide 12 by 6: \( 12 = 2 \times 6 + 0 \) (Remainder is 0)
    6. When the remainder is 0, the GCD is the last non-zero remainder, which is 6.

In this question, using the divisibility rule for 9 directly identified the greatest number among the options that divides all three numbers.