+1206 All the positive integers greater than 1 are arranged in five columns (A, B, C, D, E) as shown. Continuing the pattern, in what column will the integer 800 be written?
+36916 A C E are only odd nos.
E = 5 + 8(n) would put 797 there .....800 would be 3 more to the left in column B
+26367 All the positive integers greater than 1 are arranged in five columns (A, B, C, D, E) as shown.
Continuing the pattern, in what column will the integer 800 be written?
We rearrange:
\(\small{ \begin{array}{|cccc|cccc|cccc|cccc|cccc|c|} \hline B&C&D&E& D&C&B&A& B&C&D&E& D&C&B&A& B&C&D&E& \ldots\\ \hline 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 &10 & 11 & 12 & 13 &14 & 15 & 16 & 17 &18 & 19 & 20 & 21 & \ldots \\ \hline \end{array} }\)
\(\begin{array}{|r|r!c|c|c|c|l|} \hline \text{Position} ~&(& 1 & 2 & 3 & 4\text{ or }0& ) \\ \hline \text{Row odd} ~&(& B & C & D & E &) \\ \text{Row even} ~&(& D & C & B & A &) \\ \hline \end{array}\)
Formula:
\(\begin{array}{|rcll|} \hline \text{Row} &=& 1 + \left\lfloor \dfrac{n-2}{4} \right\rfloor \quad & \quad \left\lfloor\ldots \right\rfloor \text{Function floor or Integerpart} \\\\ \text{Position} &=& (n-1) \pmod{4} \quad & \quad \text{If the position is zero, so the position is 4}\\ \hline \end{array}\)
\(\text{Row} =\ ?, ~n = 800:\)
\(\begin{array}{|rcll|} \hline \text{Row} &=& 1 + \left\lfloor \dfrac{800-2}{4} \right\rfloor \\ &=& 1 + \left\lfloor \dfrac{798}{4} \right\rfloor \\ &=& 1 + \left\lfloor 199.5 \right\rfloor \\ &=& 1 + 199 \\ &=& 200 \quad & \quad \text{the row is even!} \\ \hline \end{array} \)
The number 800 is in row = 200. And the row is an even number.
\(\text{Position} =\ ?, ~n = 800:\)
\(\begin{array}{|rcll|} \hline \text{Position} &=& (800-1) \pmod{4} \\ &=& 799 \pmod{4} \\ &=& 3 \\ \hline \end{array}\)
\(\text{Column} =\ ?, ~\text{Row is even}:\)
\(\begin{array}{|rcll|} \hline \hline \text{Position} ~&(& 1 & 2 & \color{red}3 & 4\text{ or }0& ) \\ \hline \text{Row even} ~&(& D & C & \color{red}B & A &) \\ \hline \end{array}\)
The integer 800 will be written in column B
