To determine the largest number that divides both 255 and 579 and leaves a remainder of 3 in each case, let's follow a systematic approach:
1. Subtract the Remainder from Each Number:
The first step is to subtract the remainder (3) from both numbers. This adjustment helps us simplify the problem to finding the greatest common divisor of the new numbers.
[tex]\[
255 - 3 = 252
\][/tex]
[tex]\[
579 - 3 = 576
\][/tex]
2. Find the Greatest Common Divisor (GCD):
Now, we need to find the greatest common divisor (GCD) of the adjusted numbers, 252 and 576. The GCD is the largest number that divides both numbers without leaving a remainder.
3. Step-by-Step GCD Calculation:
Using the Euclidean algorithm to find the GCD:
- Divide 576 by 252, and get the remainder:
[tex]\[
576 \div 252 = 2 \quad \text{with a remainder of} \quad 576 - 2 \times 252 = 72
\][/tex]
- Now, take 252 and divide it by the new remainder (72):
[tex]\[
252 \div 72 = 3 \quad \text{with a remainder of} \quad 252 - 3 \times 72 = 36
\][/tex]
- Finally, divide 72 by the latest remainder (36):
[tex]\[
72 \div 36 = 2 \quad \text{with a remainder of} \quad 72 - 2 \times 36 = 0
\][/tex]
When the remainder reaches 0, the divisor at that step is the GCD. Here, the GCD is 36.
Therefore, the largest number which divides both 255 and 579 leaving a remainder of 3 is 36.
