We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
0
383
2
avatar+4221 

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

 Feb 7, 2018
 #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.

 Feb 7, 2018
 #2
avatar+22188 
+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

 Feb 7, 2018
edited by heureka  Feb 7, 2018

6 Online Users

avatar