Prove that for any 9 consecutive natural numbers, there exists one number that is prime to the other 8 numbers.

Q: Find the standard matrix for the linear transformation T: R² → R2 that reflects points about the…

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

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Q: Explain why, in any set of 11 positive integers, some two integers must have the same I's digit.

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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: How many 3-digit positive integers exist that when divided by 7 leave a remainder of 57

A: 3 digit positive integers start from 100 and ends with 999. Least positive 3 digit integer which…

Q: Compute for an arbitrary integer k

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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: Prove that 5 does not divide 5n+3 for any integer n.

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Q: In a set of 2 positive integers, how many ways it can be chosen in less than 50? 1.

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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: How many positive integers less than 1,000,000 have a digit sum of 5?

A: We have to find the number of positive integers 1,000,000 have a digit sum of 5. The number less…

Q: An integer is square-free if it is a product of distinct primes. Show that for each integer k≥ 1…

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Q: Prove that the statement is true for every positive integer n.

A: We know that the sum of a geometric series of the form: a+ar+ar2+ar3+...........+arn-1 with the…

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: Jse a direct proof to show that if n² is an odd integer, then n* is also an odd integer.

A: Given, n² is an odd integer. So, n² = 2k+1, where k is an integer. Now, n⁴ = (n²)² = (2k+1)² = 4k² +…

Q: Among all the numbers between 40000 and 90000, including the numbers 40000 and 90000, how many each…

A: The total number of numbers between 40,000 and 90,000 is 90,000-40,000+1=50,001. First calculate the…

Q: Suppose that a password is exactly six characters and each character is 1 of the 26 lowercase…

A: The number of lowercase letters is 26 and the number of integers is 10.

Q: 2. Prove that the difference of an odd integer and an even integer is odd.

A: Prove that the difference of an odd integer and an even integer is odd

Q: How Many Positive Integers ≤630 are relatively prime to 630?

A: The given number is .We have to find the number of positive integers are relatively prime to .

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

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Q: Suppose p and q are distinct prime numbers. Find the number of positive integers x ≤pq such that…

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Q: Prove this: -n is a integer; If 6| n then 18| n^2 +3n

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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: 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 distinct four-digit integers can one make from the digits 1,3,3,7,7, and 8?

A: How many distinct four-digit integers can one make from the digits 1,3,3,7,7, and 8?

Q: Show that one of the prime divisors of any three-digit composite number is less than 31 prime, or…

A: A composite integer a has a prime factor p only if p≤a that is if p≥a than there exist another prime…

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: Prove: “The sum of two even integers is even.”

A: To prove: The sum of two even integers is even. n is an even number if it can be expressed as n =…

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.