take any 11 integers. prove that there must be at least two of them that differ by an integer multiple of 10.

Q: Show that every positive integer is of the form 2q,and that every positive odd integer is of the…

A:

Q: Prove that the product of any k consecutive positive integers is an integer multiple of k!.

A:

Q: For any integer z, if 3 divides z2, then 3 also divides z.

A:

Q: Find all integers z, positive or negatives, such that z2 + z is prime.

A:

Q: rove that the negative of any even integer is even

A: Assume 2n to be an even integer. 2n = even integer -2n = negative of even integer  Let a = -2n be an…

Q: Find all integers 100 ≤ n ≤ 999 such that the last three digits of n² is same as n.

A: To find: All integers  such that the last three digits of is same as n.

Q: Compute for an arbitrary integer k

A:

Q: 2. Prove that the product of any two odd integers is an odd integer.

A: Let x and y be two odd numbers. If a and b are odd, then a and b can be written as: x=2m+1 , y=2n+1…

Q: a

A:

Q: Prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

A: The objective is to prove that 3 divides 17k+1-2*5k+3+11k+3 whenever k is a nonnegative integer

Q: Let a, b and c be integers. Prove that if a does not divide b and a|(b+c), then a does not divide c

A: Let a, b and c be integers.

Q: Find all integers x + 3 such that x – 3 is divisible by x – 3. |

A: To find all integers such that x^3-3 is divisible by x-3

Q: Let A be the set of odd integers, and let B be the set of positive integers less than 10. Find An B…

A:

Q: rove that the difference of any two odd integers is even.

A: Even numbers are of the form 2k. Odd numbers are of the form 2k+1, where k is an integer.  Let a and…

Q: Prove that the sum of any five consecutive positive integers is always a multiple of five.

A:

Q: Use two column proof and state reasons. Let x be the product of 5 distinct integers, where each of…

A: Given: Let x be the product of five unique integers, each of which is in the range of 1 to 63,…

Q: Prove that every integer n ≥ 2 can be written as a product of prime numbers.

A:

Q: if a is a positive integer less than 10, then 7a7,7a7,7a7 is divisible by 3 Select one: O True O…

A:

Q: n any positive integer n, there exist n consecutive composite integers

A: Given that given any positive integer n, show that there exist n consecutive composite integers.

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: Let a, b, c be positive integers such that a is a multiple of c and a2 + b2 is not a multiple of c.…

A: a, b, c are positive integers such that a is a multiple of c.  => there exists some integer x…

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

A:

Q: If a is any integer, is a(a + 1) even or odd?

A:

Q: Let A be the set of all positive integers whose last two digits form a number that is evenly…

A: We have the given condition as   Let us consider A be the set of all positive integers whose last…

Q: Prove that the cube of any integer in the form 9k or 9k + 1 or 9k + 8 .

A: Let n be some integer then by the Division Algorithm, n = 3p , 3p+1 or 3p+2 for some integer p