Discrete Math: Strong Math Induction

[rd_wingman]

Here is a problem form our past exam review sheet that is also on our final exam review sheet for my discrete math class. Our professor only gave us a hint on how to do it but just about everyone in the class is stuck including my self. Can someone help please?
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Compute 3^0, 3^1, 3^2, 3^3, 3^4, 3^5, 3^6, 3^7, 3^8, 3^9, and 3^10. Make a conjecture about the units digits of 3^n, where n is a positive integer. Use strong mathematical induction to prove your conjecture.

Hint from my professor:
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3^0 congruent to 1 (mod 10)
3^1 congruent to 3 (mod 10)
3^2 congruent to 9 (mod 10)
3^3 = 27 congruent to 7 (mod 10)
3^4 = 81 congruent to 1 (mod 10)
3^5 = 3^4 * 3 = 1 * 3 congruent to 3 (mod 10)
3^6 = 3^5 * 3 = 3 * 3 congruent to 9 (mod 10)
3^7 = 3^6 * 3 = 9 * 3 = 27 congruent to 7 (mod 10)

Conjecture:
3^(4k) congruent to 1 (mod 10)
3^(4k+1) congruent to 3 (mod 10)
3^(4k+2) congruent to 9 (mod 10)
3^(4k+3) congruent to 7 (mod 10)

 

[daon]

Try using the following (all arithmetic done modulo 10).

Write n=4k+r where 0<=r<4.

\(\displaystyle 3^n=3^{4k+r} = (3^4)^k \cdot 3^r = 1^k * 3^r = 3^r\)

 

[rd_wingman]

what do you mean

 

[DrMike]

what do you mean

You've made a conjecture, the next part of the question was to prove it using induction. daon is giving you a tip on how to do that.