Justify that every prime number except 2 and 3 has the form 6q + 1 or 6q + 5 for some integer q.

Q: Prove by cases that 25x² + 35x is an even integer whenever 5x-7 is an integer. Note here that the…

A: Apply the condition for 5x-7 to be an integer case by case.

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: show that if x is an integer then x2+x-41= 0 produce prime numbers

A: Here show that if x is an integer then  produce prime numbers.

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

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Q: The number of factors of prime numbers are None of the mentioned O 30 Depends on the prime number

A: No. Of factor of prime no. Means how many divisor consist of a prime no.

Q: Use a direct proof to show that if a and b are positive integers, then +> 2. Hint: Rewrite the…

A: We are given that a and b are positive integers, that means, a≥0 and b≥0 Now, the square of any…

Q: The statement (Dim K As Integer) means that, the variable K is an integer type. Select one: True…

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Q: Use indirect proof to show that, "If n is an integer and 3n+2 is even, then n is even".

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Q: Every natural number greater than 1 can be written as the products of prime numbers.

A: Prove that  every natural number greater than 1 can be expressed as product of primes or the number…

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

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Q: All prime numbers are odd.

A: We have state whether given statement is True or False. We know that  A prime number is a number…

Q: Find a number that provides a counterexample to show that a given statement is false, For all number…

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Q: Prove the statement by the method of exhaustion: For each integer n with 1 less than or equal to n…

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Q: Consider the following statement. Is there an integer that has a remainder of 2 when it is divided…

A: The statement " Is there an integer that has a remainder of 2 when it is divided by 5 and a…

Q: The integers 77 and 105 are relatively prime. صواب İhi

A: By finding the factors of given numbers you can find the solution :

Q: Show by contraposition that for any real number x if -5 <x<4 then x2+x- 20 <0

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Q: Use contradiction to prove:  if 3n + 2 is an odd integer, then n is odd.

A: consider 3n+2 as odd integer  and let  n is even  therefore, n=2k for some integer k therefore,…

Q: Show that if n is a positive integer then =n(n + 1)2"-2

A: We know the binomial theorem formula isx+yn=∑k=0nCkn xn-kyk, where Ckn is the binomial co-efficient…

Q: Show that x2 + 9 is prime.

A: To show x2 + 9 is prime.

Q: Show that for any integer n > 5, the integers n, n+2, and n +4 cannot all be primes.

A: Lets prove by contradiction Lets say integers n, n+2 and n+4 are all prime number for some integer…

Q: A. If p 2q25 is a prime number, prove that p-q is divisible by 3.

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

A: we have to prove that the statement is true for every positive integer n.…

Q: Suppose a and b are integers that divide the integer c. If a and b arerelatively prime, show that ab…

A: Since a/c   &amp;  b/c  , we know there exists integers K1 &amp; K2  ,with K1 a=K2 b=c--1Also…

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 if x is an integer then x^2+x-41= 0 produce prime numbers

A: This question is taken from the quadratic equation in which we have to find the roots of the given…

Q: Use indirect proof to show that if n is an integer and n^3 + 5 is odd, then n is even.

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Q: Every odd prime number is greater than 2.

A: Given: "Every odd prime number is greater than 2." To find: Statement in the form "If p, then q"…

Q: 3. The GCF of any two distinct prime numbers is 1. (Distinct they are not the same number) means

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Q: Show that 3 + V5 is irrational number

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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: Show that if n is an odd integer then []

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Q: show that no integer of the form n^3+1 is a prime other than 2

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Q: Prove that the number in between twin primes is always a multiple of 6.

A: A Twin prime are those numbers which are prime and having a difference of two ( 2 ) between the two…

Q: Prove that 2 - 2 · 7 + 2 · 72 – ... + 2(-7)" = (1 – (-7)n+1)/4 whenever n is a nonnegative integer.

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Q: Prove that 2^(27) - 1 is not a prime number.

A: A prime number is a natural number greater than 1 that is not a product of two smaller natural…

Q: How many integer solutions to x1+ x2+ x3+ x4 = 36 are there for which æ1 > 5, x2 > 3, x3 > 5 and æ4…

A: We have to find the number of solutions to x1+x2+x3+x4=36 for which x1≥5, x2≥3, x3≥5 &amp; x4≥6.

Q: Prove that any integer greater than or equal to 48 can be represented as the sum of nonnegative…

A: Principle of mathematical induction: For each and every natural number n, mathematical induction is…

Q: If p2q25 is a prime number, prove that p-q² is divisible by 8.

A: Given, p≥q≥5 are prime numbers, Now, p2-q2=p+qp-q. Since, p and q are prime which are grater than…

Q: The number of primes of the form n³ – 1 is 1) 1 2) 2 3) Infinite

A: To choose correct option

Q: Prove that 8 divides 9" – 1 whenever n is a natural number. For each natural number n, let P(n) be…

A: To prove that  8 divides 9^n-1 whenever n is a natural

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: Prove that no integer congruent to 3 modulo 4 can be written as the sum of two squares.

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Q: Show that 5 – V7 is an algebraic number.

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Q: Assuming every integer is either even or odd, How would I show that for any integer, the difference…

A: Assume that every integer is either even or odd. Let the integer considered say x is even. Then by…

Q: Let be a prime number (integer). Show that p2 + 5 cannot be a prime number.

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Q: Give a counterexample to the statement. The product of 2 primes is always odd.

A: Product of two primes.

Q: Every prime number is odd. True False

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Permutations and Combinations

If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.

Counting Theory

The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.