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How many two-digit positive integers are congruent to 1 (mod 3)?

 Feb 7, 2018
 #1
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3C + 10, where C =0 to 29

3*0 + 10 mod 3 = 1

3*1 + 10 mos 3 = 1

3*2 + 10 mod 3 = 1.........and so on to C=29

3*29 + 10 mod 2 = 1 and so on . So, there are 30 2-digit positive integers that sarisfy the congruence.

 Feb 7, 2018
 #2
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How many two-digit positive integers are congruent to 1 (mod 3)?

 

\(\begin{array}{|rcll|} \hline \text{ $x \equiv 1 \pmod 3$ $\\$ or $ \\ x-1 = n\cdot 3 $ } \\ \hline \end{array} \)

 

\(\begin{array}{lrcll} \text{If $x = 99$} & 99-1 &=& n\cdot 3 \\ & 98 &=& n\cdot 3 \\ & n &=& \dfrac{98}{3} \\ & n &=& 32.7 \\ & \boxed{ n = 0\ldots 32 \qquad x = 1\ldots 99 } \\ \end{array} \)

 

\(\begin{array}{lrcll} \text{If $x = 9$} & 9-1 &=& m\cdot 3 \\ & 8 &=& m\cdot 3 \\ & m &=& \dfrac{8}{3} \\ & m &=& 2.7 \\ & \boxed{ m = 0\ldots 2 \qquad x = 1\ldots 9 } \\ \end{array} \)

 

\(\begin{array}{|rcll|} \hline x = 10\ldots 99 \\ n-m &=& 33 -3 = 30 \\ \hline \end{array} \)

 

30 two-digit positive integers are congruent to 1 (mod 3)

 

\(\begin{array}{|l|rcll|} \hline 1. & 10 \\ 2. & 13 \\ 3. & 16 \\ 4. & 19 \\ 5. & 22 \\ 6. & 25 \\ 7. & 28 \\ 8. & 31 \\ 9. & 34 \\ 10. & 37 \\ 11. & 40 \\ 12. & 43 \\ 13. & 46 \\ 14. & 49 \\ 15. & 52 \\ 16. & 55 \\ 17. & 58 \\ 18. & 61 \\ 19. & 64 \\ 20. & 67 \\ 21. & 70 \\ 22. & 73 \\ 23. & 76 \\ 24. & 79 \\ 25. & 82 \\ 26. & 85 \\ 27. & 88 \\ 28. & 91 \\ 29. & 94 \\ 30. & 97 \\ \hline \end{array} \)

 

laugh

 Feb 7, 2018
edited by heureka  Feb 7, 2018

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