Combinatorics | Number Theory Answers

The formula for calculating the sum of all natural integers from 1 to n is well-known:

Sn = 1 + 2 + 3 + ... + n =

n

2 + n

2

Similary, we know about the formula for calculating the sum of the first n squares:

Qn = 1 · 1 + 2 · 2 + 3 · 3 + ... + n · n =

n

3

3

+

n

2

2

+

n

6

Now, we reduce one of the two multipliers of each product by one to get the following sum:

Mn = 0 · 1 + 1 · 2 + 2 · 3 + 3 · 4 + ... + (n − 1) · n

Find an explicit formula for calculating the sum Mn.

hannah is making a game board that has a dimension of 15 inches by 24 inches if she will use square tiles what is the largest tile she can use

For each pair of numbers find integers 𝑥 and 𝑦 such that 𝑎𝑥 + 𝑏𝑦 =gcd (𝑎,𝑏)

a) 𝑎= 91, 𝑏=10

 A code have 4 digits in a specific order, the digits are between 0-9. How many different permutations are there if one digit may only be used once?

Find a counterexample to the statement “every positive integer greater than 7 can be written as the sum of the squares of three (not necessarily unique) integers.”

Find the inverse of 77 modulo 5 by using extended Euclidean Algorithm

Step by step solution

We call a positive integer perfect if it equals the sum of its positive divisors

other than itself.

(a) Prove that 6 and 28 are perfect numbers

(b) Prove that if 2p − 1 is prime, then 2p−1

(2p − 1) is a perfect number

What is (-5 mod 4) – (-3 mod 4) congruent to?

a) 2 mod 4

b) 3 mod 4

c) 1 mod 4

d) -1 mod 4

A student must take a course on a modern language, social science, natural science and English.

There are five possible modern languages, three natural science, and four social sciences.

Every student must take the same English course.in how many ways can a student select his course of study.