To solve the problem of finding the largest number that divides 66, 76, and 96 leaving a remainder of 6 in each case, we need to follow a systematic approach.
1. Understand the Remainder Condition:
When a number divides another number leaving a remainder of 6, it means that if we subtract 6 from that number, the resulting number should be divisible by our desired divisor.
2. Subtract the Remainder:
Let's subtract 6 from each of the given numbers:
- For 66: [tex]\( 66 - 6 = 60 \)[/tex]
- For 76: [tex]\( 76 - 6 = 70 \)[/tex]
- For 96: [tex]\( 96 - 6 = 90 \)[/tex]
3. Determine Divisibility:
Now, we need to find the largest number that can divide 60, 70, and 90 exactly. This largest number is known as the greatest common divisor (GCD).
4. Find the Greatest Common Divisor (GCD):
To find the GCD of multiple numbers, we can use the following properties of the GCD:
- The GCD of two numbers is the largest number that can divide both of them without leaving a remainder.
- The GCD of multiple numbers can be found by iteratively finding the GCD of two numbers at a time.
5. Calculate the GCD Step-by-Step:
- First, find the GCD of 60 and 70:
[tex]\( \text{GCD}(60, 70) = 10 \)[/tex]
- Next, use this result to find the GCD with the third number, 90:
[tex]\( \text{GCD}(10, 90) = 10 \)[/tex]
6. Conclusion:
The largest number that divides 66, 76, and 96 leaving a remainder of 6 each time is 10.
Therefore, the answer is 10.
