any set of 201 integers must contain at least 21 elements that all have the same last digit.

Q: How many integers from 1 to 700 (inclusive) are divisible by the factors of 70 greater than 1?

A:

Q: Suppose 35 cars start at a car race. In how many ways can the top 3 cars finish the race?

A: The information provided in the question are as follows :-There are  cars at a car race.

Q: If A is the set of all whole numbers less than 20 that are divisible by, 4, and B is the set of all…

A: A∪B is the union of set A and set B. This set contains all the elements in set A or set B. The Venn…

Q: Show that a number with 30 digits cannot have more than 100 prime factors.  

A: Prime factor: A factor of a number which is a prime number. For example: consider a number 26. The…

Q: Name all integers from -100 to 100 that are congruent to -1 modulo 25!

A:

Q: Prove that a number in the form 6k+5,” what I mean is a number in the form of 6 times some integer…

A:

Q: How many divisors does (1000000! + 1) have that are less than 1000000?

A:

Q: Use any method, list all the three digit numbers from the set {3,4,5,6} where there are no repeating…

A:

Q: Determine the number of six-character passwords that can be formed using two letters from A to F and…

A: From A to F there are 6 letters. We can pick 2 letters from 6 letters in 6C2=15 waysIn 1 to 9 there…

Q: Let Sbe the set of all (nonnegative) 4-digit integer. How many integers from start with 1 and end…

A: let S be the set of all 4-digit integer. how many integers from start 1 and end with 9

Q: (3) Prove that if a cube of a number is divisible by 17, then the number is divisible by 17.

A: We have to prove that if cube of a any number is divisible by 17 then the number itself is divisible…

Q: How many positive integers less than 10000 are there which contain at least one 2 or at least one 3…

A: The objective is to evaluate the number of positive integers less that 10000 that contain at least…

Q: How many elements are there in the greatest conjugacy class of 5 ?S

A:

Q: noosing a positive integer from 100 to 999, what is the poSSibility of choosing a positive integer…

A:

Q: what two consective odd integers have a product of 63?

A: Find two odd integers.

Q: 7. Show that for every natural number n2 1, the integer ) + () +*(;) + - +) +22 +... +2" is…

A: Using binomial theorem:  n0+2n1+22n2....2nnn=n01n·20+n11n-1·21+n21n-2·22....nn1n-n·2n…

Q: How many integers between 1000 and 9999 inclusive are not divisible by 5?

A: The number of integers from 1 and 9999 which are divisible by 5 is = [9999 ÷ 5] = [1,999.8] = 1999…

Q: Suppose 59 cars start at a car race. In how many ways can the top 3 cars finish the race? ... The…

A: There is total 59 cars.

Q: Prove by contradiction that there is no largest composite number.

A:

Q: In how many integers greater than 0 and smaller than 10000 digit 5 appears exactly once?

A:

Q: write a program to evaluate even numbers for [1,10],and [11,20]respectively in matlab

A: Even numbers for [1,10] is given below-

Q: Assume that all four-digit integers (numbers from and including 1000 to and including 9999) are…

A:

Q: True/False: Every integer greater than 1 is divisible by a prime number. True False

A:

Q: Let the binary operation *

A: Given,          a*b=a+2b where a and b are real numbers

Q: How many real four digit numbers (they can not start with zero) can be formed from digits 0, 1, 2,…

A: Given digits is 0,1,2,3,4,5,6Total no.of digits = 7

Q: How many positive integers with 6 or less digits are odd or divisible by 5 and do not contain any…

A: We can solve this problem by considering two cases:

Q: Show what the divisors of 120 are and what they add up to.

A: Divisors of 120=1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120

Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.