15.A certain number when successively divided by 3 and 5 leaves reminder 1 and 2. what is the reminder, if the same number be divided by 15?

To solve this problem, we need to work backwards from the given conditions.

1. Start with the second condition: the number leaves a remainder of 2 when divided by 5. The smallest such number is 2 (since 0 is not considered), but it could also be 7, 12, 17, etc. In general, we can express this as 5n + 2, where n is any non-negative integer.

2. Now consider the first condition: when this number is divided by 3, it leaves a remainder of 1. Substituting 5n + 2 into this condition, we get (5n + 2) mod 3 = 1.

3. Solve this equation for n. The smallest positive solution is n = 1, which gives us the number 7 (since 5*1 + 2 = 7).

4. Finally, divide this number by 15 and find the remainder. 7 mod 15 = 7.

So, the remainder when the number is divided by 15 is 7.

To solve this problem, we need to work backwards from the given conditions.

1. Start with the second condition: the number leaves a remainder of 2 when divided by 5. The smallest such number is 2 (since 0 is not considered), but it could also be 7, 12, 17, etc. In general, we can express this as 5n + 2, where n is any non-negative integer.

2. Now consider the first condition: when this number is divided by 3, it leaves a remainder of 1. Substituting 5n + 2 into this condition, we get (5n + 2) mod 3 = 1.

3. To find the smallest such number, we can start with n = 0 and increase it until we find a value that satisfies the equation. Doing this, we find that n = 1 works, giving us a number of 5*1 + 2 = 7.

4. Finally, we need to find the remainder when this number is divided by 15. Doing this, we find that 7 mod 15 = 7.

So, the remainder when the number is divided by 15 is 7.