Consider the following addition problem: 3P+4P+PP+PP = RQ2; where P, Q and R are different digits. What is…
Consider the following addition problem: 3P+4P+PP+PP = RQ2; where P, Q and R are different digits. What is the arithmetic mean of all such possible sums?
102
120
202
220
Solution
Writing the two-digit numbers: $(30+P) + (40+P) + (10P+P) + (10P+P) = 100R + 10Q + 2$, i.e. $24P + 70 = 100R + 10Q + 2$. The unit digit 2 of the sum arises when 4P ends in 2, so P = 3 or P = 8. If P = 3, the sum is $24 \times 3 + 70 = 142$. If P = 8, the sum is $24 \times 8 + 70 = 262$. Arithmetic mean = $(142 + 262)/2 = 202$.