Finding the largest number that divides 62,132,237 to leave the same remainder.

To solve this problem, we need to use the concept of the remainder theorem. The remainder theorem states that if we divide a number by a divisor, then the remainder will be the same if we divide the number by any multiple of the divisor.

For example, if we divide 62,132,237 by 7, we get a remainder of 5. If we divide the same number by 14 (which is a multiple of 7), we will still get a remainder of 5.

Using this concept, we can start by finding the remainder when 62,132,237 is divided by a small number and then check if the remainder is the same when we divide the number by any multiple of that small number.

Let's start by checking if 62,132,237 is divisible by 2, 3, 5, 7, and 10.

- Divisible by 2: The last digit of 62,132,237 is odd, so it is not divisible by 2.
- Divisible by 3: The sum of the digits of 62,132,237 is 26, which is not divisible by 3, so it is not divisible by 3.
- Divisible by 5: The last digit of 62,132,237 is not 0 or 5, so it is not divisible by 5.
- Divisible by 7: Let's use long division to check if 62,132,237 is divisible by 7.

_________
7 | 62,132,237
56,000,000
__________
6,132,237
5,600,000
__________
532,237
490,000
_______
42,237
35,000
______
7,237
7,000
_____
237

The remainder is 237, which is not 0, so 62,132,237 is not divisible by 7.

- Divisible by 10: The last digit of 62,132,237 is 7, which is not 0, so it is not divisible by 10.

Since 7 is the largest number that divides 62,132,237 to leave a remainder, we need to check if the remainder is the same when we divide the number by any multiple of 7.

Let's check if the remainder is the same when we divide 62,132,237 by 14, 21, 28, and 35.

- Divisible by 14: The remainder when 62,132,237 is divided by 14 is 5, which is the same as the remainder when it is divided by 7.
- Divisible by 21: The remainder when 62,132,237 is divided by 21 is 10, which is not the same as the remainder when it is divided by 7.
- Divisible by 28: The remainder when 62,132,237 is divided by 28 is 1, which is not the same as the remainder when it is divided by 7.
- Divisible by 35: The remainder when 62,132,237 is divided by 35 is