+0  
 
0
43
2
avatar+1704 

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

tertre  Feb 7, 2018
Sort: 

2+0 Answers

 #1
avatar
+1

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.

Guest Feb 7, 2018
 #2
avatar+18956 
+3

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

heureka  Feb 7, 2018
edited by heureka  Feb 7, 2018

13 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details