A number n is divisible by 12 if it satisfies ∃k ∈Z(n = 12k) Give a direct proof that, if n is divisible by 12 and m is even, then 18m −3n is divisible by 12.

Q: Calculate C(12,5) by hand. As C(n,r) = n!/(n-r)!r! identify n and r, and write out every number in…

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Q: Show that 7 divides 29(52k) + 17(25k-2) for any integer k ≥ 1.

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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: Prove that a number in the form 6k+5,” what I mean is a number in the form of 6 times some integer…

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Q: 5. (12 points) Find how many positive integers between 1 and 99, inclusive, have the following…

A: The objective of this question is to find the number of positive integers between 1 and 99 that are…

Q: If x is divisible by 5 or y is divisible by 5, then xy is divisible by 5.

A: Let x is divisible by 5. Then x=5u for some natural number u.

Q: 5. (12 points) Find how many positive integers between 1 and 99, inclusive, have the following…

A: The objective of this question is to find the number of positive integers between 1 and 99 that…

Q: divisible by 3 for

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Q: Let n be a number with 3 digits. (i.e 100sns 999) Show that if the sum of digits of n is divisible…

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Q: How many positive integers between 5 and 31 (a) are divisible by 3? (b) are divisible by 4? (C) are…

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Q: 6) Let n be a positive integer such that 7 | (52n-1 +22n-1). Prove that 7 || (52n+1+22n+¹). (Hint:…

A: Since, given that  52n-1+ 22n-1  is divisible by 7 .  We have to prove that  52n+1+22n+1 is also…

Q: Disprove  For any integer n, n2 + 5 is not divisible by 5.

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Q: Prove by mathematical induction that for all integers n≥1, 2n+² +3²n+¹ is divisible by 7.

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Q: How do we proof that 2logn = n?

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Q: 14. Prove that 32n +7 is divisible by 8. 15. Prove that 7"-2" is divisible by 5. 16. Prove that n³ -…

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Q: Use mathematical induction to prove that 3n+ 7n−2 is divisible by 8 for every non-negative integer…

A: The given statement is, Pn=3n+7n-2 Let us assume that the statement is true for n=1 P1=31+71-2=8 As…

Q: For all integers n> 0, 32n – 1 is divisible by 8.

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Q: Prove by mathematical induction that for all integers n 2 0: a 5" + 2 x 11" is divisible by 3, b 33n…

A: (a) Case 1 : n=1 51+2×111 = 5+2×11= 5+22= 27 As 27 is a multiple of 3 , hence the result holds for…

Q: let n be an integer solution to the equation n2+an-b=0 (n>7) if the representation of a in base n is…

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Q: 9. 10. What is t divisors?

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