Proof by Induction: Divisibility by 17

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SUMMARY

The discussion focuses on proving that the expression 3 * 5^(2n+1) + 2^(3n+1) is divisible by 17 using mathematical induction. The user successfully established the base case for n=1 and proceeded to the inductive step by assuming the expression holds for k and attempting to prove it for k+1. The proof was completed by demonstrating that the sum of two numbers, both divisible by 17, remains divisible by 17, confirming the validity of the expression.

PREREQUISITES
  • Understanding of mathematical induction
  • Familiarity with divisibility rules
  • Basic algebraic manipulation skills
  • Knowledge of exponential functions
NEXT STEPS
  • Study mathematical induction techniques in detail
  • Explore divisibility proofs involving modular arithmetic
  • Learn about exponential growth and its properties
  • Practice similar proofs with different bases and exponents
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Mathematics students, educators, and anyone interested in number theory or proof techniques will benefit from this discussion.

guropalica
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Proof by induction that 3 * 5^2n+1) + 2^3n+1 is divisible by 17!
Thanks in advance guys
 
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In your expression, there is an ")" without an "(" --> 3 * 5^2n+1) + 2^3n+1

so it is not clear what do to!
 
You also need to try yourself and let us know where you're stuck. Then we can help.
 
I proved the base case n=1, and then I try doing the step case assuming that it satisfies for any k, then I try proving it by k+1.
I got sth like 24 * (3 * 5^(2k+1)) + 7 * 2^(3k +1) Now I'm stuck can't continue, I don't have any ideas :/
btw the initial equation is 3 * 5^(2n+1) + 2^(3n+1) !
 
Got it now :) let's denote the initial statement as K + L, so we have sth like 24l + 7k = 7(k+l) + 17k, sum of two numbers divisible by 17 is divisible by 17, anyway thx
 

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