Since the next prime after 7 is​ 11, the smallest composite number whose prime factors are all greater than​ 7, is ▼     Since this is ▼   less greater than​ 100, there are no ▼   prime composite ​numbers, from 2 to​ 100, whose prime factors are all greater than 7.​ Therefore, ▼   every at least one composite number from 2 to 100 must be a multiple of some prime number less than or equal to 7.

Q: How many integers between 100 and 400 contain the digot 2?

A:

Q: are there any even numbers greater than 2 which are prime? why or why not?

A: We will find the solution in the following step

Q: NOT ALL THE NUMBERS ARE ON HERE!!

A: Solution-: Given data: 18,80,83,295,332,489,503,513,530,544,570,606,793,840,912,980 We want to find…

Q: n^2 +1 is not divisible by 3 for every integer n.

A: Any integer n can be written in the form of: n=3a+b, where b = 0 or 1 or 2 Case 1: If b=0, n2+1=…

Q: A number is called “quadary” if each digit is either 0, 1, 2, or 3. How many different 5-digit…

A: Given that a number quadary if each digit is either 0,1,2 or 3. To find the number of  different…

Q: Find the number of positive integers X smaller than 12849 which have exactly 2022 positive divisors

A: Theory : Total no of divisor of a number, If its prime factorization, given by ∏i=1hpiai Where p is…

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: + 220 + 330 + 440 + 550 + 660 (

A:

Q: How many natural numbers that are less or equal to 1000 and that are not multiples of 3, 5, or 7…

A: A number is said to be divisible by other number if on dividing the number  with it given remainder…

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

A:

Q: Every even integer greater than 2 is sum of two primes? Give your views on this?

A: We know that 2 is even prime. And all other primes are odd.

Q: If both p and p^2 +8 are prime numbers then show that p^3+4 is also prime number

A:

Q: If the first even integer is 4, then what will be next three consecutive even integers ? O 5, 14, 10…

A: In the question it is asked to represent next three consecutive even integers after even integer 4.

Q: Prove by cases For any integer n, n^2 + 5 is not divisible

A:

Q: Show that if seven integers are selected from {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, there must be…

A:

Q: Find how many natural numbers n ≤ 1000 are not divisible by any of 2, 3 without repetitions

A:

Q: Find the number of positive integers not exceeding 104 that are not divisible by 5 or by 7.

A: In this Question we have to find the number of positive Integer between 0 and 104 which is not…

Q: How many positive integers less than 1000 are divisible by 5? How many positive integers less than…

A: We will answer the first three questions as per our guideline. Please resubmit the other question…

Q: find all possible three-digit numbers that are of the form 75_ and are divisible by both 3 and 2

A: Since the number is divisible by 2 and 3, it must be divisible by 6.

Q: If O is a natural number then O^2 + 2O + 1 is odd.

A: If O is a natural number then we need to check whether  O^2 + 2O + 1 is odd. The result is given…

Q: How many 5-digit numbers (including leading 0s)are there with no digit appearing exactly two times?

A:

Q: Every even integer greater than 2 is sum of two primes? Give your views on this?

A:

Q: How many 5- digit numbers can be formed using the digits 0,1,2,3,4,5,6,7,8,9, if repetitions of…

A: Givendigits = 0,1,2,3,4,5,6,7,8,9No.of digits =10

Q: Find all whole#s that leave remainder 3 upon division by 5?

A: The solution is given below:

Q: How many integers between 100 and 999 inclusive are multiples of 13?

A:

Q: Show that given any set of 5 natural numbers, there are 3 numbers whose sum is divisible by 3.

A: Let q∈N. Note that, for every q there exists some x∈N and some v∈0,1,2 such that q=3x+v. Now, let…

Q: How many consecutive zeroes are there at the end of 100! (100 factorial). For example, 12! -…

A: How many consecutive zeroes are there at the end of 100! We know that…

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

A:

Q: Show that there are no prime triples other than 3, 5, 7.

A:

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

A:

Q: prove that Every positive integer n can be written as n =( 2^a)(5^b)c, where c is not divisible by 2…

A: Fundamental Theorem of Arithmetic: Every positive integer (except n=1) can be uniquely expressed as…