Sushant took six AIMCATs and the total score that he obtained in each AIMCAT was a distinct integer between 121 to 130, both inclusive. If the average of his AIMCAT scores at the end of every AIMCAT was an integer, and he scored 127 in the sixth test, how much did he score in the fifth test?

To solve this problem, we need to understand that the average of Sushant's AIMCAT scores at the end of every test was an integer. This means that the sum of his scores after each test was a multiple of the number of tests he had taken up to that point.

Let's denote the scores of the six tests as A, B, C, D, E, and F (in chronological order). We know that F = 127.

The sum of the scores from the first test to the sixth test is A + B + C + D + E + F. Since the average of these six scores is an integer, this sum must be a multiple of 6.

The possible sums of six distinct integers between 121 and 130 (inclusive) are 741, 747, 753, 759, 765, 771, and 777. Among these, only 747 and 765 are multiples of 6.

If the total sum was 747, then the sum of the scores from the first test to the fifth test would be 747 - F = 747 - 127 = 620. However, this is not a multiple of 5, which contradicts the fact that the average score after the fifth test was an integer.

Therefore, the total sum must have been 765. This means that the sum of the scores from the first test to the fifth test was 765 - F = 765 - 127 = 638. Since this is a multiple of 5, it is consistent with the fact that the average score after the fifth test was an integer.

To find the score of the fifth test (E), we need to subtract the sum of the scores of the first four tests from 638. The sum of the four smallest possible scores (121, 122, 123, and 124) is 490. Therefore, E = 638 - 490 = 148.

However, this is not possible because the score of each test was a distinct integer between 121 and 130. Therefore, the sum of the scores of the first four tests must have been larger than 490.

The sum of the four smallest possible scores excluding 121 (122, 123, 124, and 125) is 494. Therefore, E = 638 - 494 = 144.

This is still not possible. Therefore, the sum of the scores of the first four tests must have been larger than 494.

The sum of the four smallest possible scores excluding 121 and 122 (123, 124, 125, and 126) is 498. Therefore, E = 638 - 498 = 140.

This is still not possible. Therefore, the sum of the scores of the first four tests must have been larger than 498.

The sum of the four smallest possible scores excluding 121, 122, and 123 (124, 125, 126, and 127) is 502. Therefore, E = 638 - 502 = 136.

This is a valid score, so Sushant must have scored 136 in the fifth test.