Suppose the product of 22 integers is equal to 1. Prove that their sum is not equal to 0.

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: Find the least positive integer that yields the remainders 1, 3, and 5 when divided by 5, 7, and 9,…

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Q: Suppose that you know that a golfer plays the first hole of a golf course with an infinite number of…

A: To prove that the golfer plays every hole on the course when given that a golfer plays the first…

Q: Without using any theorems from the book, prove the following statement: The sum of any two odd…

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Q: Show with the help of Fermat’s little theorem that if n is a positive integer, then 42 divides n7 −…

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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: The sum of any two integers of the from 4k+ is always: a) prime b) zero c) Even

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Q: Show that the sum of the cubes of the first n positive integers is (n(n + 1)/2)^2

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Q: Find the smallest positive integer that leaves a remainder of 1 when divided by 2, a remainder of 2…

A: Let N be such a number N=10a+9:  N+1 = 10a+10 N=9b+8: N+1 = 9b+9 N=8c+7: N+1 = 8c+8 N=7d+6 and so on…

Q: Consider all positive integers that are mueiples of 20 and that are less than or equal to 300. What…

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Q: of the first 150 positive integers?

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Q: If x is an odd integer then 5x – 7 is an even integer.

A: 5 and x are odd integers. Multiplication of two odd integers is always odd. Hence, 5x is an odd…

Q: does there exists   ______such that if n is divided by 5 the remainder is 2 and  if   ?

A: We have to check whether there exist n such that if n is divided by 5 then remainder is 2. Let if n…

Q: Suppose we have L-shaped tiles made up of three squares: Prove that for all n > 1, it's possible to…

A: Suppose we have L shaped tiles Prove that for all n>=1 it's possible to lay these tiles down to…

Q: Show that if any eight positive integers are chosen, two of them will have the same remainder when…

A: Pigeonhole Principle- If n items are assigned to m containers, and m<n,then at least one…

Q: Every even integer greater than 2 is sum of two primes? Give your views on this?

A: We know that 2 is even prime. And all other primes are odd.

Q: If m times the nth term of an AP is equal to n times its nth term, show that the (m + n) th term of…

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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 2^10 + 5^12 is a composite number.

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Q: Define an odd number and show that the product of odd even numbers is also even.

A: We need to first define an odd number. And layer prove product of an odd and an even number is…

Q: Prove that 3:8 is equivalent to 12:32

A: According to the question, we have to prove that the ratio 3:8 is equivalent to 12:32. A ratio is a…

Q: Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the remainder…

A: we have to Show that for all integers m and n, if the remainder is 2 when m is divided by 4 and the…

Q: Prove that the sum of the squares of two odd integers cannot be the square of an integer.

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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: If the sum of any two integers is even, then so is their difference.

A: As per our guidelines we are supposed to answer only one question. Kindly repost other question as…

Q: Prove that the product of two odd numbers is odd.

A: We first define a general odd number. Lets say that n is an integer. When we multiply any number by…

Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: prove that the one's digit of any integer,n, is the remainder when one divides n by 10.

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Q: Suppose that all the entries in A are integers and det A = 1. Explain why all the entries in A–1 are…

A: Inverse of a matrix A is calculated by,

Q: prove that it ņ is an 3n +2 is odd then integer and is add

A: This is related to Number System

Q: If the sum of two counting numbers is an even number, then the product of the two counting numbers…

A: ANSWER:- 1) From the given statement, If the sum of two counting numbers is an even counting number,…

Q: Find an integer that leaves a remainder of 2 whan divided by either 3 or 5, but that is divisible by…

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Q: if two even integers are multiflied, then the product is also even

A: Ans. True

Q: There is no integer that has a reciprocal that is an integer.

A: As per the instructions, we will be answering only the first question. For (32), Notice that the…

Q: Prove 8 divides 34h+1+52h+1 whenever h is a positive integer

A: We need to prove that 8 divides 34h+1+52h+1 whenever h is a positive integer by Mathematical…

Q: The sum of any two integers of the from 4k + 1 is always: Even O prime Zero Odd

A: If a number n is not divisible by 2 then it is an odd number.  If a number n divisible by 2 then it…

Q: Prove (NOT BY EXAMPLE) that for every nonzero integer, its square is positive.

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Q: Prove by contradiction that 2020 cannot be written as a sum of 3 odd integers.

A: The objective is to prove  that 2020 cannot be written as a sum of 3 odd integers by the…

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

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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: 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: Determine whether the statement is true or false. Justify your answer with a proof counterexample,…

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

A: We need to prove by mathematical induction that, 3 divides 17k+1-2*5k+3+11k+3 whenever k is a…

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: the sum of any positive integers that is less than 30

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Q: True or false, for any integer x, x^2, - x will always result in an even number

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Q: Let x and y be positive integers. When is x + y3 a prime number? When is x+y a composite number?…

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