The numbers 4121, 4973 and 6464 leave the same remainder x in each case when divided by the greatest number y. the value of (2y-x) is;352336332432

1. Break Down the Problem

We need to find the values of $y$ (the greatest number) and $x$ (the common remainder) such that the following holds true:

• $4121 \equiv x \, (\text{mod } y)$
• $4973 \equiv x \, (\text{mod } y)$
• $6464 \equiv x \, (\text{mod } y)$

From this, we can conclude that:

• The differences between the numbers must be divisible by $y$:
  • $4973 - 4121$
  • $6464 - 4973$
  • $6464 - 4121$

2. Relevant Concepts

For any two numbers $a$ and $b$, if $a \equiv x \, (\text{mod } y)$ and $b \equiv x \, (\text{mod } y)$, then $a - b$ is divisible by $y$.

3. Analysis and Detail

First, calculate the differences:

1. $4973 - 4121 = 852$
2. $6464 - 4973 = 1491$
3. $6464 - 4121 = 2343$

Now, we need to find the greatest common divisor (GCD) of these differences:

• $\text{gcd}(852, 1491, 2343)$

Let's find the GCD step by step.

Step 3.1: Finding the GCD of 852 and 1491

Using the Euclidean algorithm:

• $1491 \mod 852 = 639$
• $852 \mod 639 = 213$
• $639 \mod 213 = 0$

So, $\text{gcd}(852, 1491) = 213$.

Step 3.2: Finding the GCD of 213 and 2343

Using the Euclidean algorithm:

• $2343 \mod 213 = 0$

So, $\text{gcd}(213, 2343) = 213$.

Thus, $y = 213$.

4. Verify and Summarize

Now we need to calculate the remainder $x$:

• Using one of the original numbers, say $4121$:
  • $4121 \mod 213 = 4121 - (19 \times 213) = 4121 - 4047 = 74$

Thus, $x = 74$.

Now, calculate $2y - x$:

• $2y - x = 2 \times 213 - 74 = 426 - 74 = 352$.

Final Answer

The value of $2y - x$ is $352$.