Dividing a number by 68
To find the remainder when a number is divided by 68, we are given that the quotient is 269 and the remainder is 0. Let's assume the number is represented by 'x'.

Dividing the number by 68:
x = 68 * 269 + 0
x = 18292

Therefore, the number is 18292 when divided by 68.

Dividing the same number by 67:
Now, we need to find the remainder when this number (18292) is divided by 67.

To find the remainder, we can use the concept of congruence. Congruence is a mathematical relationship that represents the equality of remainders when two numbers are divided by the same divisor.

We can write the congruence as:
x ≡ r (mod 67)

Where 'x' is the number, 'r' is the remainder, and 'mod' represents the modulus or divisor.

Applying the concept of congruence:
x ≡ r (mod 67)
18292 ≡ r (mod 67)

To solve this congruence, we need to find a number that is congruent to 18292 modulo 67. In other words, we need to find a number that leaves the same remainder as 18292 when divided by 67.

Calculating the remainder:
To simplify the calculation, we can express 18292 as the sum of multiples of 67 and the remainder 'r'.

18292 = 67 * 273 + r
18292 = 18291 + r

From this equation, we can see that the remainder 'r' is 1.

Therefore, when the number 18292 is divided by 67, the remainder is 1.

Summary:
- When the number is divided by 68, the quotient is 269 and the remainder is 0.
- The number is 18292 when divided by 68.
- To find the remainder when the same number is divided by 67, we use the concept of congruence.
- The remainder is found by expressing the number as a sum of multiples of 67 and the remainder 'r'.
- The remainder 'r' is 1 when the number 18292 is divided by 67.

269/ 67 * 68/67
= 1 * 1 = 1. I think.