Let P = 5⁵ × 15¹⁵ × 25²⁵ × 35³⁵ and Q = 10¹⁰ × 20²⁰ × 30³⁰ × 40⁴⁰. What is the number of consecutive zeros at the end of the sum (P + Q)?

P contains no factor of 2 (all bases 5, 15, 25, 35 are odd), so P is odd. Q contains many factors of 2 and 5. The number of trailing zeros in Q: powers of 2 in Q = 10 + 40 + 30 + 120 = 200; powers of 5 = 10 + 20 + 30 + 40 = 100. So Q ends in exactly 100 zeros (min of 200, 100). P ends in many 5s but no 2s, so P has 0 trailing zeros. Sum P + Q: the last digit of P is 5 (odd), Q ends in 100 zeros. P's trailing zeros count: powers of 5 in P = 5+15+50+35 = 105, powers of 2 = 0, so P has 0 trailing zeros — last digit is 5. Then P + Q ends in 5, having zero trailing zeros. Wait — re-examining: Q ends in 100 zeros means Q = ...0000 (100 zeros), and P ends in ...5. Adding gives last digit 5, hence zero trailing zeros. Answer: Zero.