Sure, let's go through the problem step-by-step.
You are asked to find the greatest number which divides 97 and 116 leaving a remainder of 2 in each case. Let's break this down.
1. Understanding the remainder condition:
We are given that when 97 is divided by some number [tex]\(x\)[/tex], it leaves a remainder of 2. Mathematically, this can be expressed as:
[tex]\[ 97 \equiv 2 \ (\text{mod} \ x) \][/tex]
This implies:
[tex]\[ 97 = kx + 2 \][/tex]
for some integer [tex]\(k\)[/tex].
Similarly, for 116:
[tex]\[ 116 \equiv 2 \ (\text{mod} \ x) \][/tex]
This implies:
[tex]\[ 116 = mx + 2 \][/tex]
for some integer [tex]\(m\)[/tex].
2. Simplify the equations:
Let's subtract 2 from both 97 and 116 to simplify:
[tex]\[ 97 - 2 = kx \][/tex]
[tex]\[ 116 - 2 = mx \][/tex]
Which means:
[tex]\[ 95 = kx \][/tex]
[tex]\[ 114 = mx \][/tex]
3. Greatest Common Divisor (GCD):
Now we need to find the greatest number [tex]\(x\)[/tex] that divides both 95 and 114. This means finding the greatest common divisor (GCD) of 95 and 114.
4. Finding the GCD:
We can find the GCD using the Euclidean algorithm:
- Step 1: Divide 114 by 95 and find the remainder.
[tex]\[ 114 \div 95 = 1 \ \text{R} \ 19 \][/tex]
Here, the remainder is 19.
- Step 2: Now, divide 95 by the remainder from the previous step, which is 19.
[tex]\[ 95 \div 19 = 5 \ \text{R} \ 0 \][/tex]
Since the remainder is now 0, we stop here.
The last non-zero remainder we obtained is the GCD, which is 19.
So, the greatest number that divides both 97 and 116 leaving a remainder of 2 is 19.
