To find the number of positive integers, a, such that 2027 divided by a leaves a remainder of 7, we can use a straightforward approach.
First, let's understand the concept of remainder when dividing two numbers. When a number, n, is divided by another number, a, the remainder is the leftover value after dividing n by a.
In this case, we are given that 2027 divided by a leaves a remainder of 7. Mathematically, this can be represented as:
2027 = a × q + 7
Here, q represents the quotient of the division.
To find the number of positive integers, a, that satisfy this condition, we can iterate through all the possible values of a and check if the equation is satisfied.
We know that the remainder can range from 0 to (a - 1). However, since we are specifically looking for a remainder of 7, we can ignore all the values of a less than 7, as they will never satisfy the equation.
Therefore, we can start iterating from a = 7 onwards and increment a by 1 for each iteration. For each value of a, we can check if the equation 2027 = (a × q) + 7 is satisfied.
If it is satisfied, then a is a valid positive integer. We can count the number of valid integers, a, until we find the first value where the equation is no longer satisfied.
To summarize the steps:
1. Start with a = 7.
2. Compute the quotient, q, using the equation q = (2027 - 7) / a.
3. If q is an integer, increment the count of valid integers, and continue to the next iteration with a + 1.
4. Repeat steps 2-3 until the equation is no longer satisfied.
5. The count of valid integers, a, is the number of positive integers that satisfy the given condition.
Implementing this process will help us find the number of positive integers, a, that satisfy the equation 2027 = a × q + 7 and have a remainder of 7 when divided by a.