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Math Help - Stuck on Proofs question

  1. #1
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    Stuck on Proofs question

    This is as far as I have gotten. Any recommendations would be much appreciated.


    Proposition. For each natural number n, 3 divides 5^n-2^n.

    Proof. We will use the Principle of Mathematical Induction. We let P(n) be “3 divides 5^n-2^n.”

    For the basis step, we must prove that P(1) is true. We note that 5^1-2^1=3=3(1) and 1 is an integer. Therefore, we know that 3 divides 5^1-2^1 as 3(1) = 5^1-2^1, and consequently we can conclude that P(1) is true.

    For the inductive step, we prove that for all natural numbers k, if P(k) is true, then P(k+1) is true. So let k be a natural number and assume that P(k) is true. That is assume that,

    3|(5^k-2^k ).

    Therefore, we know that there exists an integer m such that

    (5^k-2^k )=3m or 5^k=3m+2^k. (1)

    In order to prove that P(k+1) is true, we must show that 3|5^(k+1)-2^(k+1). Multiplying both sides of equation (1) by 5 gives us,

    5^k∙5=(3m+2^k )5
    5^(k+1)=15m+2^k∙5

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  2. #2
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    Quote Originally Posted by eg37se View Post
    This is as far as I have gotten. Any recommendations would be much appreciated.


    Proposition. For each natural number n, 3 divides 5^n-2^n.

    Proof. We will use the Principle of Mathematical Induction. We let P(n) be “3 divides 5^n-2^n.”

    For the basis step, we must prove that P(1) is true. We note that 5^1-2^1=3=3(1) and 1 is an integer. Therefore, we know that 3 divides 5^1-2^1 as 3(1) = 5^1-2^1, and consequently we can conclude that P(1) is true.

    For the inductive step, we prove that for all natural numbers k, if P(k) is true, then P(k+1) is true. So let k be a natural number and assume that P(k) is true. That is assume that,

    3|(5^k-2^k ).

    Therefore, we know that there exists an integer m such that

    (5^k-2^k )=3m or 5^k=3m+2^k. (1)

    In order to prove that P(k+1) is true, we must show that 3|5^(k+1)-2^(k+1). Multiplying both sides of equation (1) by 5 gives us,

    5^k∙5=(3m+2^k )5
    5^(k+1)=15m+2^k∙5
    Yuo should note that 5^{k+1} - 2^{k+1} = 5 \cdot 5^k - 2 \cdot 2^k = 5 \left( 5^k - 2^k \right) + 3 \cdot 2^k and each term is divisible by 3.
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  3. #3
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    Thanks a bunch
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