A) 0
B) 1
C) 2
D) 63
Correct Answer: A
Solution :
| [a] We have |
| \[{{7}^{9}}+{{9}^{7}}={{(8-1)}^{9}}+{{(8+1)}^{7}}={{(1+8)}^{7}}-{{(1-8)}^{9}}\] |
| \[=[1+{{\,}^{7}}{{C}_{1}}8+{{\,}^{7}}{{C}_{2}}{{8}^{2}}+...+{{\,}^{7}}{{C}_{7}}\,{{8}^{7}}]\] |
| \[-[1-{{\,}^{9}}{{C}_{1}}8+{{\,}^{9}}{{C}_{2}}{{8}^{2}}-....-{{\,}^{9}}{{C}_{9}}{{8}^{9}}]\] |
| \[={{\,}^{7}}{{C}_{1}}8+{{\,}^{9}}{{C}_{1}}8+[{{\,}^{7}}{{C}_{2}}+{{\,}^{7}}{{C}_{3}}.8+...-{{\,}^{9}}{{C}_{2}}+\,\] |
| \[^{9}{{C}_{3}}.8-...]{{8}^{2}}\] |
| \[=8(7+9)+64\,k=8.16+64\,k=64\,q.\] |
| Where \[q=k+2\] |
| Thus, \[{{7}^{9}}+{{9}^{7}}\] is divisible by 64. |
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