24. Suppose that I choose six of the first ten positive integers. Prove that I must have chosen two numbers such that one is a divisor of the other. [HINT: Write each of the 10 numbers as a power of 2 multiplied by an odd number, then use the pigeonhole principle.]

Q: How do I prove that if a, b, and c are integers with c not equal to 0 then ac = bc => a = b?

A:

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A: We have to find how many 5’s does the prime factorization of x + y contain.

Q: Prove that if n is a +ive integer,

A: Given that, M xddxnf(x);p = (-1)npnF(p) We can prove this by mathematical induction.

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: 2. Prove by mathematical induCHon that १) के) 22 32 42 2n is Hrue For all integers nz 2

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Q: Prove the statement. "For any integer n, nº +4 is not divisible by 8." by Contradiction.

A: Given: n be any integer. To prove: n3+4 is not divisible by 8.

Q: 4) Prove that the product of any three consecutive integers is divisible by 6.

A: We have to show that product of any three consecutive integers is divisible by 6

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Q: Show by contraposition that for all integers n^2, if  is odd, then n + 1 is even.

A:

Q: . Is there an integer that has a remainder of 2 when it is divided by 5 and a remainder of 3 when…

A: Consider an integer x that has a remainder of 2 when it is divided by 5 Thus x=2 mod 5 0r x=5k+2 It…

Q: Use the ertended pigeonhole principle to show that there is at least 15 ways to choose 4 integers…

A: Extended Pigeonhole Principle:      If there are n pigeons and m pigeonholes and n>m. Then, there…

Q: Let A be the set of positive integers that are less than or equal to 40 and are multiples of 4 and…

A: Given: Let A be the set of positive integers that are less than or equal to 40 and are multiples of…

Q: Suppose a set of 8 numbers are selected from the set: {1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13;…

A: It is given that 8 numbers are selected from the set 1,2,3,4,5,6,7,8,9,10,11,12,13,14.

Q: 6. Use contradiction to prove that, for all integers k > 1, 2/k+1+ 2/K+2

A:

Q: Prove that 2^10 + 5^12 is a composite number.

A:

Q: i need the last one, thank you

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Q: 2. Prove by contraposition: For all integers n, if (3n +4) is even, then n is even.

A: Contrapositive of 'if p then q' is if not q the not p'

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: Prove that the last two digits of 2" cannot be 02 and the last three digits cannot be 108. Find all…

A: All these problems can be solved using modular arithmetic. We presented the detailed solution in the…

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Q: Suppose n and m are integers. Prove by contrapositive: If (n + m) is an even integer and (nm) is an…

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Q: Discrete

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Q: 1- Show that if a is an integer, then 3 divides a^3-a. with details plz

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Q: Suppose thatk and n 2k are fixed positive integer. Justify the identity k+1 jk

A: to prove ∑j=knjk=n+1k+1above equation can be written as S2(j,k)=S2(j-1,k-1)+k.S2(j-1,k)count the…

Q: 21. How many positive integers x are there such that 3x has 3 digits and 4x has four digits?

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

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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   &  b/c  , we know there exists integers K1 & K2  ,with K1 a=K2 b=c--1Also…

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Q: 2. Classify as integer or non-integer. • 28 • -58 • 7/16 • 7/21 • 21/7

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Q: Prove that for all integers a, b, and c, if a2 + b2 = c2, then at least one of a and b is even.

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Q: Suppose we randomly select a subset of positive integers. How large must the subset be, if we want…

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Q: 2. Show that if a]b then a'|b² for all integers a and b (Symbols V, 3, A, V, V,A,→,-)

A:

Q: Prove that when you select n +1 numbers from {1,2, ... 2n} two of them must be consecutive (that is,…

A: Use inductive argument and Pigeon hole principles

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Q: prove by contrapositive : for all n,m integers if nm is even then n is even or m is even

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Q: Prove that 11 divides 3* 5k + 42k+1 + 33k+2 – 5 * 72k for every k non-negative integer

A: Thanks for the question :)And your upvote will be really appreciable ;)

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Q: * 10. Is the following proposition true or false? Justify your conclusion. For each integer n, n is…

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Q: 1. Use Direct Proof to prove that the product of two consecutive integers is divisible by 2.

A: We have to prove that product of two consecutive integers is divisible by 2.

Q: 1- The total number of positive integers that can be formed from the digits 1,2,3 and 4 if no digit…

A: NOTE- AS PER POLICY I  HAVE CALCULATED FIRST MAIN QUESTION ONLY. Here Given 4 digits 1,2,3,4

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Q: Let A be the set of five-digit numbers where only the digits 1, 2 and 3 are included. For example…

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Q: Prove that if a is a positive integer, then 4 does not divide a2 + 5.

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Q: Let x and y be positive integers such that the prime factorization of x has exactly four 5’s and…

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Q: Show that 1 (N, <) is well-ordered. 1 2 (Z, <) is not well-ordered. 3 (R, <) is not well-ordered.

A: Please send questions as different questions. We cannot do more than 3 at a time.   A relation ≤ on…