Question

When 27 is divided by 6, the remainder is 3, and when 27 is divided by 7, the remainder is 6. What is the next greatest positive integer that has a remainder of 3 when divided by 6 and a remainder of 6 when divided by 7?

1) 39
2) 48
3) 51
4) 69
5) 90

Answer 1

The next greatest positive integer that has a remainder of 3 when divided by 6 and a remainder of 6 when divided by 7 is 69, as it is the only option that satisfies both conditions.

Step-by-step explanation:

To find the next greatest positive integer that has a remainder of 3 when divided by 6 and a remainder of 6 when divided by 7, we need to look for a number that satisfies both conditions. These types of problems are commonly solved using methods from number theory, such as the Chinese Remainder Theorem or systematic checking. However, since we have a list of options, we can simply check each one.

Let's analyze the options provided:

1. 39 divided by 6 gives a quotient of 6 and a remainder of 3, and 39 divided by 7 gives a quotient of 5 and a remainder of 4. So, 39 doesn't satisfy the second condition.
2. 48 divided by 6 gives a quotient of 8 and a remainder of 0, and 48 divided by 7 gives a quotient of 6 and a remainder of 6. While the remainder when divided by 7 is correct, it's not when divided by 6.
3. 51 divided by 6 gives a quotient of 8 and a remainder of 3, and 51 divided by 7 gives a quotient of 7 and a remainder of 2. So, 51 doesn't satisfy the second condition either.
4. 69 divided by 6 gives a quotient of 11 and a remainder of 3, and 69 divided by 7 gives a quotient of 9 and a remainder of 6. This number satisfies both conditions and is the correct answer.
5. 90 divided by 6 gives a quotient of 15 and a remainder of 0, and 90 divided by 7 gives a quotient of 12 and a remainder of 6. Again, only the remainder when divided by 7 is correct.

Therefore, the next greatest positive integer after 27 that has a remainder of 3 when divided by 6 and a remainder of 6 when divided by 7 is 69.