A natural number greater than 1 that has only itself and 1 as factors is called a/an number.

Q: The number 6 has four divisors -- namely: 1, 2, 3, and 6. List all of the numbers less than 20 that…

A: Given that: Divisors of number 6 = 1, 2, 3, 6

Q: Find all primes that are the difference of the fourth powers of two integers.

A: Let the two integers are x and y and n is prime so we have  n=(x4-y4)       (suppose x>y)…

Q: How many positive integers less than 10000 are there which contain at least one 7 or at least one 9…

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Q: Prove that the last two digits of 2" cannot be 02 and the last three digits cannot be 108.

A: Note: As per our company guidelines we are supposed to answer the first question only. Kindly ask…

Q: Compute how many different ways there are to color the faces of a cube so that three faces are red,…

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Q: 1. How many 5 digit positive integers are divisible by 5 and have at least one 6 as their digit?

A: A number is divisible by 5 if the last digit is either 5 or 0. We have to find the number of 5 digit…

Q: Compute the sum of the first 60 positive integers that are exactly divisible by 4.

A: The numbers divisible by 4 is:

Q: what

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Q: How many six-digit positive integers contain exactly one one, two twos and three treor? How many…

A: First note that here six digit integer can start only with 1 or 2. If it starts with 1 the remaining…

Q: Compute C7,2. (Enter an exact number.)

A: A combination is a selection of all or part of a set of objects, without follow any order in which…

Q: Two distinct numbers are selected from the set (1,2,3,4,...,36,37) so that the sum of the remaining…

A: Given two distinct numbers are selected from the set 1,2,3,4,…,36,37 and The sum of remaining 35…

Q: How many four digit positive integers are there that contain the digit 3 and are divisible by 5?

A: A number is divisible by 5, if the unit digit place contains the number 0 or 5. Let m choices to be…

Q: How many five-digit numbers are there for which each digit is either equal to both adjacent ones, or…

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Q: Any integer greater than 1 that has other positive factors besides 1 and itself is also known as O…

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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: Show that every positive integer can be written as the product of a square (possibly 1) and a…

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Q: How many 3-digit numbers can be formed from the digits 2, 3, 5, 6, and 9, which are divisible by 5,…

A: Given information- We have given the digits- 2, 3, 5, 6, and 9. We have to form the 3 digits numbers…

Q: a) How many 3-digit positive integers are there with odd digits?

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Q: Prove that 2^10 + 5^12 is a composite number.

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Q: What is the largest number in the row starting "1 4 ..." of Pascal's triangle?   What is the sum…

A: To find what is the largest number in the row starting "1 4 ..." of Pascal's triangle? To find what…

Q: Prove that there is no triangular number which is one less than a multiple of 11.

A: Introduction: A triangular number is a positive integer that is the average of two integers. For…

Q: Find all three-digit numbers with different digits with the property that those numbers are…

A: Let N=100a+10b+c represent all the three-digit numbers, for some digits, a, b and c. Then,…

Q: Prove that a three-digit number is a multiple of 9 whenever the sum of the digits is a multiple of…

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Q: Prove that an odd number multiplied by an odd number is odd.

A: The objective is to prove that an odd number multiplied by an odd number is odd.

Q: Suppose you must use 1x“-tile and 10 x-tiles. You only have up to 15 1-til use. How many of the…

A: Introduction: A rectangle is a quadrilateral with equal angles and opposite sides that are equal and…

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

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

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Q: How many different integers must be selected to guarantee that at least 5 of them would have the…

A: Dividing any integer by 3, can have remainder 0, 1 or 2. Therefore, possibilities for remainder are…

Q: Show that, selecting 91 positive integers guarantees that there are at least 10 integers with the…

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Q: Compute the sum of the first 50 positive integers that are exactly divisible by 5.

A: To determine sum of the first 50 positive integers that are exactly divisible by 5.

Q: Compute P5,4. (Enter an exact number.)

A: Obtain the value of P5,4.  The value of P5,4 is obtained below:   Permutation: A permutation is an…

Q: show that every integer greater than 11 is the sum of two composite integers

A: We need to prove that  every integer greater than 11 is the sum of two composite integers.

Q: Suppose a password must have at least 8 characters, but no more than 12 characters, made up of…

A: Given that, Suppose a password must have at least 8 characters, but no more than 12 characters, made…

Q: Prove that a four-digit number is a multiple of 8 whenever the number formed by the first three…

A: The solution is given by using the expanded form of number as follows

Q: The total number of positive integers that can be formed from the digits 1,2,3 and 4 if no digit is…

A: Number of possible arrangement can be calculated either by combination or permutation methods.…

Q: Find all three-digit numbers with different digits such that those three-digit numbers are divisible…

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Q: How many three-digit odd integers can be formed using the digits 0, 1, 4, 5, 6, 7, and 9 only once?

A: Permutation: A permutation of a set is defined as an arrangement of its members into a sequence or…

Q: How many 6-permutations of [15] have their digits in increasing order?

A: Permutation is a technique to obtain the possible arrangements of listed items in a linear order.…

Q: How many ways are there to permute the elements of the multiset {1,1,2,2,3,4,5,6} so the first and…

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Q: ) How many,8-distinct/digit numbers can be formed if the numbers áre amultiple of/5 and contain the…

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Q: How many 5 digit positive integers are divisible by 5 and have at least one 6 as their digit?

A: A number is divisible by 5, if it has a 0 or 5 in the unit digit place. Let m choices to be made…

Q: Show that if seven integers are selected from the first 10 positive integers, there must be at least…

A: We can solve this question by pigeonhole principle

Q: How large a subset of the integers from 1 to 50 must be chosen to guarantee that one of the numbers…

A: We have to determine How large a subset of integers from 1 to 50 must be chosen to guarantee that…

Q: 5!, called 5_______ , is the product of all positive integers from_________ down through________ .…

A: For any positive integer n, n! is the product of all the positive integers  from 1 to n. That is,…

Q: Consider 4 digit numbers with each digit from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} .  How many such…

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Q: A natural number greater than 1 that is divisible by a number other than itself and 1 is called a/an…

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Q: How many four-digit positive integers both start and end in even digits?

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Q: How many 3-digit numbers can be formed from the prime numbers from 0 to 10 if the repetition of the…

A: We need to find how many 3 - digit numbers can be formed from the prime numbers from  0  to  10  if…

Q: Prove that the sum of two consecutive odd integers is divisible by 4.

A: Let the two consecutive integers be n and n+2 Then we need to prove that (n)+(n+2) is divisible by 4…