Skip to main content

2008 AMC 10B Problem 8

Problem 8 of 25EasierNumber Theory

A class collects $50\$50 to buy flowers for a classmate who is in the hospital. Roses cost $3\$3 each, and carnations cost $2\$2 each. No other flowers are to be used. How many different bouquets could be purchased for exactly $50?\$50?

Answer choices

Show solution

Solution

If rr roses and cc carnations are bought, then 3r+2c=50.3r+2c=50. Because 2c2c and 5050 are even, 3r3r must be even, so rr is even. Also 3r≤50,3r\le 50, so r≤16.r\le 16. The even values r=0,2,4,…,16r=0,2,4,\ldots,16 each give a valid c,c, which is 99 bouquets. Thus, the correct answer is C.
AoPS wiki

Tagged: Diophantine Equation · parity

More practice