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: 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: 5. Show that 2 32 3" 31 3" for all positive integers n.

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

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

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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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