R and S are 2-digit positive integers. R is a multiple of 5, and S is a multiple of 9. What is the least possible value of the product RS?

Rationale

90 is the least possible value of the product RS.

To find the least product of two-digit integers R and S, where R is a multiple of 5 and S is a multiple of 9, we consider the smallest two-digit multiples of each. R must be at least 10 and S must be at least 18, leading to a minimum product of 10 x 9 = 90.

A (50) YOUR ANSWER

The value of 50 cannot be the product of R and S because the smallest two-digit multiple of 5 is 10, and the smallest two-digit multiple of 9 is 18. Thus, the lowest product, starting from these values, is 10 x 9 = 90, which is greater than 50.

B (90) CORRECT

This choice represents the product of the smallest two-digit multiple of 5 (which is 10) and the smallest two-digit multiple of 9 (which is 18). The calculation yields 10 x 9 = 90, confirming that this is indeed the least product achievable under the given conditions.

C (95) YOUR ANSWER

A product of 95 cannot be achieved by multiplying two-digit integers R and S that meet the criteria of being multiples of 5 and 9, respectively. The closest combinations—like 10 x 9 or 15 x 6—do not yield 95, reinforcing that 90 is the minimum product.

D (145) YOUR ANSWER

The product of 145 does not align with the multiples required for R and S. The pairings of two-digit multiples of 5 and 9 will either yield a product lower than 145 or not reach this value at all, given the constraints of their respective ranges.

E (180) YOUR ANSWER

While 180 is a valid product of two-digit multiples (e.g., 15 x 12), it exceeds the minimum possible product derived from the smallest valid multiples of 5 and 9. Therefore, 90 remains the least product possible.

Conclusion

For the integers R and S, the minimum possible product fulfilling the conditions of R being a multiple of 5 and S being a multiple of 9 is 90. This product stems from the smallest two-digit multiples of each factor. Other options either fall short of this minimum or exceed it, underscoring the significance of calculating from the lowest valid multiples.

R and S are 2-digit positive integers. R is a multiple of 5, and S is a multiple of 9. What is the least possible value of the product RS?

90 is the least possible value of the product RS.

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