Of the positive integers that are less than 25, how many are equal to the sum of a positive multiple of 4 and a positive multiple of 5?

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) YOUR ANSWER

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) YOUR ANSWER

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) YOUR ANSWER

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) CORRECT

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) YOUR ANSWER

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.

Of the positive integers that are less than 25, how many are equal to the sum of a positive multiple of 4 and a positive multiple of 5?

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.

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