To find the positive integer n that satisfies these conditions, we can express the conditions mathematically:
- When n is divided by 3, the remainder is 2: n ≡ 2 (mod 3)
- When n is divided by 5, the remainder is 1: n ≡ 1 (mod 5)
Now, we start by listing the possible values for n based on the first condition:
- If we take n = 2 (since 2 is the smallest positive integer satisfying n ≡ 2 (mod 3)), we find that this does not satisfy n ≡ 1 (mod 5).
- If we take n = 5 (the next integer after 2 that would fit the mod 5 condition), checking gives us a remainder of 2 with mod 3, so we reject this.
- If we take n = 8, we again do not have a fit. Continuing this, we find n = 11, which satisfies both conditions:
- 11 mod 3 = 2
- 11 mod 5 = 1
In fact, since the conditions are cyclic, any integer of the form n = 11 + 15k (where k is a non-negative integer) will also satisfy both conditions. Thus, the sequence of integers which meet the requirements includes 11, 26, 41, and so on.
In summary, the positive integer n that satisfies the conditions of the problem statement is 11 and it can also be represented in the general form of n = 11 + 15k.