Determine all natural numbers divisible by 8 whose sum of digits is less than 10 and the product of digits is equal to 12.

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

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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: Distinguish between finite and infinite?

A: Finite means that the quantity is limited . It does not mean that it can be calculated on the piece…

Q: pair of these integers with a sum equal to 9.

A: Let us take set under consideration {1,2,3,4,5,6,7,8}Possible pairs {{1,8}, {2,7}, {3,6}, {4,5}}…

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

A: Given that  A natural number greater than 1 that has only itself and 1 as factors is called…

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 telephone numbers have no repeated digits?

A: Given, A telephone number consists of seven digits which is combination of {0,1,2,3,4,5,6,7,8,9}.

Q: A company wants to have 6-digit phone extensions with the first digit not being zero. How many…

A: Given, No. of digit in phone extension = 6 First no- cannot be  0 Therefore no. of ways it can be…

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

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Q: Suppose 51 cars start at a car race. In how many ways can the top 3 cars finish the race? The number…

A: Solution: From the given information, number of cars start at a car race is 51.

Q: All composite number can represent as product of primes numbers

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Q: How many 5–digit (decimal) integers are there with none of the digits repeated?

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Q: Derive an expression for the sum of the first n positive even integers.

A: We Use the A.P series. Find The sum of the first n positive even integer.

Q: Set C is the set of positive two-digit even numbers less than 56 that are divisible by 5.

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Q: Determine the identity element of the set N={1,2,3,..} of positive integers with addition.

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Q: Prove that for any 9 consecutive natural numbers, there exists one number that is prime to the other…

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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: What is the largest number x such that x<200 and x has exactly 4 factors?

A: Factors:    Number that are multiplied together to get a product                              Factor…

Q: "Every positive integer can be written as the sum of the square of 3 positive integers." prove…

A: Given: "Every positive integer can be written as the sum of the square of 3 positive integers." To…

Q: The set of numbers {0, 1, 2, 3, 4, ...} is known as the negative integers. whole numbers. positive…

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

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Q: The sum of all positive integers less than 300 which are not divisible by 3.

A: Sum of n positive integer is given by, Sn=nn+12

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

A: Find two odd integers.

Q: How many pairs of ordered numbers exist such that sum , product and quotient are the same?

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Q: Prove that the product of an even number and an odd number is even.

A: Here, objective is to prove that the product of an even number &amp; an odd number is even

Q: Show A and Show B are two of the most popular television shows of all time. The number of episodes…

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Q: In a large math conference with 2022 participants, each person shakes hands with some number of…

A: For 'n' number of participants the maximum number of handshakes is n(n-1)/2.

Q: Suppose that 6 machines can complete a certain task in 9 days. If there are 18 machines, how many…

A: Concept: If number of machine increase then number of work completion day will decrease . Hence Both…

Q: Prove: The product of 2 numbers is equal to the product of their least common multiple and their…

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Q: How many positive integers not exceeding 3000 are divisible by exactly two of 4, 10, 12? (Steps…

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Q: Possible products whose factors multiply to −2 are?

A: To find the solutions of XYZ = -2 in integers

Q: Find the average of the first 50 natural numbers is……………….

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Q: 4. ( it as linear combination of both numbers. Find the greatest common divisor of the numbers 1520…

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Q: Determine the smallest integer that can be expressed as a sum of two primes in three different ways.

A: The number is 22 and all others smallest integer can not be written as sum of two primes in three…

Q: Prove that a natural number (an-1an-2a1ao)10 is divisible by 9 if and only if the sum of the digits…

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Q: A set of 32761 pigeons flies home, each to one of 14 gigantic pigeonholes. What is the smallest…

A: Given that, A set of 32761 pigeons flies home, each to one of 14 gigantic pigeonholes.

Q: Show that every odd integer is the difference between two consecutive perfect squares.

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Q: 4 There is an infinite number of ways to represent 1 on the number line. In the space below vrika au…

A: We can express 1 in an infinite number of ways Have to write the four expressions multiplying by 1…

Q: Find the number of positive integers ≤ 1000 that are not divisible by 5, 13, or 31?

A: .  Symbol of Greatest Integer Function  To Find: Number of positive integers ≤ 1000 that are not…

Q: Show that in any collection of 7 or more natural numbers there are ALWAYS TWO numbers whose sum or…

A: In the given problem we have to show that in any collection of 7 or more natural numbers there are…

Q: Find five consecutive odd integers such that when the fifth is subtracted from eigt times the…

A: We're going to solve

Q: Put a positive whole number in each of the four boxes of this table that satisfies the rule. For…

A: Prime number is a positive integer having exactly 2 factors. for example 7 is a odd prime number as…

Q: Let ab5c be all four-digit multiples of number 36 with different digits. If someone found 7 numbers…

A: Let ab5c be all four digit multiples number 36 with different digits. As 36= 4x9, so the number ab5c…

Q: 9. Prove that the greatest common divisor of two positive integers divides their least common…

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Q: Suppose that all the pages are black and white, but each group of 25 consecutivepages (1-25, 26-50,…

A: To determine the number of ways for the 125 pages to assign to the 5 printers. Given that, the total…

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

A: We need to find the number of positive integers both start and end in even digits.

Q: What 2 nonnegative numbers with a sum of 23 have the largest possible product?

A: Given information: The two non negative numbers with a sum of 23.  Calculation: Consider x and y are…