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: Use the factorization A = PDP Ak= - 7 40 - 2 11 = 45 1 1 1 01 30 to compute A I -1 5 1 - 4 1 where k…

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Q: Let c, a, B be nonzero real numbers such that c + B. Find a given that 132n 3 8 ac? c-B" Türkçe: c,…

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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: 16. Given an odd integer a, establish that a² + (a + 2)²+(a + 4)² + 1 is divisible by 12.

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Q: 32. Prove that 3 divides n³ +2n whenever n is a positive integer.

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

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Q: 3. Let a and b be positive integers. Express the positive greatest common divisor of a and b in…

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

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

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

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Q: Use the factorization A = PDP to compute AK, where k represents an arbitrary positive integer. - 7…

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

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Q: 5) Let n be a positive integer such that 7 | (8-1). Prove that 7 | (8=*1 - 1). (Hint: Rewrite 8*1…

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

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