Rationale
There are 11 positive integers less than 25 that are equal to the sum of a positive multiple of 4 and a positive multiple of 5.
To find the positive integers less than 25 that can be expressed as the sum of a positive multiple of 4 and a positive multiple of 5, we start by identifying the combinations of these multiples. The valid sums derived from multiples of 4 and 5 yield 11 unique integers below 25.
A) 2
The number 2 cannot be formed by adding a positive multiple of 4 and a positive multiple of 5 since the smallest positive multiple of either is 4 or 5, respectively. Thus, 2 is not a valid sum.
B) 5
Although 5 is a positive integer, it cannot be expressed as the sum of a positive multiple of 4 and a positive multiple of 5, as the smallest positive multiple of 4 is 4. Therefore, 5 is not a valid sum either.
C) 10
While 10 is a valid sum (for example, 5 + 5), it does not represent all sums of positive multiples of 4 and 5 below 25. The total count of such sums includes integers beyond just 10.
D) 11
This choice correctly identifies the total quantity of positive integers less than 25 that can be formed by adding positive multiples of 4 and 5. The valid sums include integers like 9, 10, 14, 15, 18, 19, and others, which add up to 11 unique integers.
E) 22
While 22 is a positive integer and can be derived from the sums, this option does not reflect the count of unique integers that can be expressed as the sum of positive multiples of 4 and 5. Thus, it does not answer the question correctly.
Conclusion
The task of identifying integers less than 25 that can be formed as sums of positive multiples of 4 and 5 leads to a total of 11 unique integers. Understanding the combinations of these multiples allows us to recognize that the total count aligns precisely with the correct answer, confirming D as the accurate choice.
