Answer :
To solve this problem, we need to find the greatest number that divides the numbers 62, 132, and 237 while leaving the same remainder in each case.
Here is a step-by-step solution:
1. Understand the Problem:
If a number, say [tex]\( x \)[/tex], leaves the same remainder when dividing the numbers 62, 132, and 237, then the differences between these numbers, when divided by [tex]\( x \)[/tex], should also have no remainder. In other words, [tex]\( x \)[/tex] should divide the differences between these numbers completely.
2. Calculate the Differences:
- Difference between 132 and 62:
[tex]\( 132 - 62 = 70 \)[/tex]
- Difference between 237 and 132:
[tex]\( 237 - 132 = 105 \)[/tex]
- Difference between 237 and 62:
[tex]\( 237 - 62 = 175 \)[/tex]
3. Find the Greatest Common Divisor (GCD):
The greatest number that divides these differences completely will be the greatest number that divides 62, 132, and 237 leaving the same remainder. To find this number, we need to calculate the GCD of these differences:
- First, find the GCD of 70 and 105:
The GCD of 70 and 105 is 35.
- Next, find the GCD of this result (35) and 175:
The GCD of 35 and 175 is also 35.
4. Conclusion:
The greatest number that divides 62, 132, and 237 leaving the same remainder is 35.
Here is a step-by-step solution:
1. Understand the Problem:
If a number, say [tex]\( x \)[/tex], leaves the same remainder when dividing the numbers 62, 132, and 237, then the differences between these numbers, when divided by [tex]\( x \)[/tex], should also have no remainder. In other words, [tex]\( x \)[/tex] should divide the differences between these numbers completely.
2. Calculate the Differences:
- Difference between 132 and 62:
[tex]\( 132 - 62 = 70 \)[/tex]
- Difference between 237 and 132:
[tex]\( 237 - 132 = 105 \)[/tex]
- Difference between 237 and 62:
[tex]\( 237 - 62 = 175 \)[/tex]
3. Find the Greatest Common Divisor (GCD):
The greatest number that divides these differences completely will be the greatest number that divides 62, 132, and 237 leaving the same remainder. To find this number, we need to calculate the GCD of these differences:
- First, find the GCD of 70 and 105:
The GCD of 70 and 105 is 35.
- Next, find the GCD of this result (35) and 175:
The GCD of 35 and 175 is also 35.
4. Conclusion:
The greatest number that divides 62, 132, and 237 leaving the same remainder is 35.
