Combinatorics | Number Theory Answers

Prove for all integers a and b, if a mod 8 = 5 and b mod 8 = 3 then ab mod 8 = 7.

In how many ways can five science books and six mathematics book can placed on a shelf if they must be placed alternately

Which of the options in this scenario are correct? A car dealership has 10 BMWs , 5 KIAs , and 15 AUDIs. The ratio 3:2 compares.

Qn 1. An integer N has digital representation a1a2a3. Moreover,

ˆ None of the digits a1, a2, or a3 and none of the numbers with digital

representation a1a2, a1a3, or a2a3 is divisible by 3.

ˆ N is odd.

ˆ N is divisible by 9.

ˆ a1 ≥ a2 ≥ a3.

Determine all possible numbers N.

given that n is a positive even integer, 5n + 4 will always be divisible

A. 5n

B.5

C. 2

D. 4

Three neon lights colored red, blue and green flash at different time intervals. The red light flashes after every 18 seconds and the green light after every 15 seconds. If all the three lights flash together at 8:00 am, how many times will all three lights flash together by 9:30 am.

Please show your full steps answer.

Find the last digit of the sum

0! + 2! + 4! +...+ 2010! + 2012!

Please show full steps answer.

Twelve cowboys sit in a circle around a bonfire. Each observes that his age (viewed as integer) is the average of the ages of his left and right neighbors. What is the sum of their ages?

Please show your answer with all complete steps.

a) i) Give an inductive formula for the sum of the first n odd numbers:

1 + 3 + 5 + ... + 2n -1

Show your induction process.

ii) Use the proof by mathematical induction to prove the correctness of your

inductive formula in i) above.