Finding the Largest Number with Specific Remainders

The question asks for the largest number that divides three given numbers, 627, 15630, and 3128, leaving specific remainders: 2, 5, and 3, respectively.

Let the largest number be $N$. According to the problem statement:

• When 627 is divided by $N$, the remainder is 2. This means $627 - 2$ is perfectly divisible by $N$.
• When 15630 is divided by $N$, the remainder is 5. This means $15630 - 5$ is perfectly divisible by $N$.
• When 3128 is divided by $N$, the remainder is 3. This means $3128 - 3$ is perfectly divisible by $N$.

So, $N$ must be a common divisor of the numbers obtained by subtracting the remainders:

• $627 - 2 = 625$
• $15630 - 5 = 15625$
• $3128 - 3 = 3125$

Since we are looking for the largest number that divides these three numbers, we need to find the Highest Common Factor (HCF) or Greatest Common Divisor (GCD) of 625, 15625, and 3125.

Calculating the HCF (Greatest Common Divisor)

To find the HCF of 625, 15625, and 3125, we can use the prime factorization method.

Let's find the prime factors of each number:

• Prime factorization of 625: \[625 = 5 \times 125 = 5 \times 5 \times 25 = 5 \times 5 \times 5 \times 5 = 5^4\]
• Prime factorization of 15625: \[15625 = 5 \times 3125 = 5 \times 5 \times 625 = 5 \times 5 \times 5^4 = 5^6\]
• Prime factorization of 3125: \[3125 = 5 \times 625 = 5 \times 5^4 = 5^5\]

Now we identify the common prime factors and their lowest powers. The only common prime factor is 5. The powers of 5 are $5^4$, $5^6$, and $5^5$. The lowest power is $5^4$.

Therefore, the HCF of 625, 15625, and 3125 is $5^4$.

Calculating the value of $5^4$:

So, the largest number that divides 627, 15630, and 3128 leaving remainders 2, 5, and 3 respectively is 625.

Summary of Calculation

HCF (625, 15625, 3125) = Lowest power of common prime factors = \(5^4 = 625\).

Final Answer Determination

The largest number is 625.

Revision Table: HCF with Remainders

Additional Information: Understanding HCF

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of two or more non-zero integers is the largest positive integer that divides each of the integers without leaving a remainder.

Methods to find HCF include:

• Prime Factorization Method:
  1. Find the prime factorization of each number.
  2. Identify the common prime factors.
  3. For each common prime factor, take the lowest power that appears in any of the factorizations.
  4. Multiply these lowest powers together to get the HCF.
• Euclidean Algorithm:
  1. To find the HCF of two numbers, divide the larger number by the smaller number 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 HCF.
  4. For more than two numbers, find the HCF of two numbers, then find the HCF of the result and the next number, and so on.

In this problem, since all numbers are powers of 5, prime factorization was the most straightforward method.