Problems on numbers

Question

Q1. Let x₁,x₂,x₃,...,x₁₀₀ be hundred integers such that the sum of any five integers is equal to 20. Then

1. the largest x_i equals 5;
2. the smallest x_i equals 3;
3. x₁₇=x₈₃; 4.none of the foregoing statement is true.

Q2. The smallest positive integer n with 24 divisors (where 1 and n are also considered as divisors of n) is

1. 420;
2. 240;
3. 360;
4. 480.

I could not figure out the first question.But for the second one, the answer says it is 360. So I tried to solve it in this way that is:

360 = 3*3*2*2*2*5

So, the total no of divisors =

                         = 64

So I took the factors of the 360 and each and every factor is a divisor of 360. And I used the combinations to create new divisors for the 360. So I want to know what is wrong with this method.

Answer 1

For the first one, you should prove that all the numbers are equal. Find two sets of five that only have one element different, then those two elements must be equal. Keep going. For the second, look up the divisor function. The number of factors of a number depends on the exponents of the primes in its factorization.