AMC 10 mock test problems pending
Q1. Let N=1×2×3×⋯⋯×n, where n is a positive integer. Find the largest possible value of n for which N is not divisible by some two-digit number. S1: First thought - for n>=5 10 will divide each factorial so what is the question even asking? But reading it carefully, we note that it's asking something else. Find the largest n! which is not divisible by at least one 2-digit number. 4! is not divisible by any(hence at least one) 2 digit number but it's not the largest. Largest such number could possibly be 99! . But it is divisible by all 2-digit numbers including 99. 98! is divisible by all except 99 but 11*9 = 99 so not correct. 97! is also divisible by 99 and 98(49*2). 96! is not divisible by 97 which is a prime. Hence 96!. Q2. Real numbers a and b satisfy a+b=7 and ab=−3. What is the value of a^3/b^2 + a +b^3/a^2 + b? S2: a^3/b^2 + a +b^3/a^2 + b = (a + b) + (a^5 + b^5)/(a^2b^2) We know the values of ab, a+b so we just need to focus on a^5 + b^5 which we can pro...