Problem Solving

Determine the sum of the integers among the first 1000 positive integers, which are not divisible by 4 or are not divisible by 9. (This is not an exclusive.) Let R be the region consisting of points (x,y) in the Cartesian plane satisfying both the absolute value of x - the absolute value of y ≤ 1 and the absolute value of y ≤. Sketch the region R and find it's area. How many positive integers n are there that are exact divisors of at least one of 10^40 and 20^30 ? Please see attached.

Pigeon hole theorem (I believe)

Okay, here's the problem; Let A be any set of twenty integers chosen from the arithmetic progression 1, 4, 7, ...,100. Prove that there must be two distinct integers in A whose sum is 104.

Formula used to determine the number of different ways to deal ...

Show that 2(2^n-1 - 1) is the formula used to determine the number of different ways to deal n distinct playing cards to two players where each player gets at least one card. I want to allow the possibility of giving a different number of cards to each player.

Problem Solving: Proofs & Sequences

Please help me with these three problems. I am a novice at writing proofs and deriving formulas so, I am not sure if I am on the right track. Questions (also attached): Practice problem 1 Fn is the Fibonacci sequence (f0 = 0, f1 = 1, fn+1 = fn + fn-1). By considering examples, determine a formula for the following expressions, and then verify the formula. a. f0 + f2 + f4 + …+f2n b. f0 - f1 + f2 - f3 + …+(-1)n fn --------------------------------------------- Practice problem 3 By observation, derive a formula for (n 0) + (n 1)2 + (n 2)^2 +…+(n n)2^n = the summation n where k=0 (n k)2^k. Verify your formula. ( ) are being used to express n chose zero, n chose ... click for more

Sum of series

Determine the sum of the integers among the first 1000 positive integers which are not divisible by 4 or are not divisible by 9. (This is not an exclusive or)