Final answer:
To find how many distinct positive integers n can be expressed as n=ab where a and b are between 1 and 10, we list all possible products and count only the distinct products. After listing and removing duplicates, the result is 55 distinct integers, so the answer is d. 55.
Step-by-step explanation:
To determine how many distinct positive integers n can be expressed as n = ab for integers such that 1 ≤ a ≤ 10 and 1 ≤ b ≤ 10, we should consider all the possible products of integers a and b within the given range.
If we list down all the products for a and b where a and b are both between 1 and 10, we will have a 10x10 grid of pairs (a, b). However, pairs like (2,3) and (3,2) will result in the same product, so we must be careful to count each distinct product only once.
Let's create a list by multiplying each pair a and b:
- a=1, b=1 to 10 => 1, 2, 3, ..., 10 (10 distinct numbers)
- a=2, b=1 to 10 => 2, 4, 6, ..., 20 (9 distinct numbers since 2 is already counted)
- a=3, b=1 to 10 => 3, 6, 9, ..., 30 (7 distinct numbers since 3, 6, 9 are already counted)
- Continuing this pattern, we add the distinct counts for each a.
After completing this process, we find there are 55 distinct positive integers that can be expressed as the product of two integers between 1 and 10.
Therefore, the correct answer is: d. 55.