| Statement I \[[{{(m+1)}^{7}}-{{m}^{7}}-1]\] is divisible by 7 for each\[m\in N.\] |
| Statement II \[[{{m}^{7}}-m]\] is divisible by 7 for each \[m\in N.\] |
A) Statement I is true and Statement II is true. Statement II is the correct explanation for Statement I
B) Statement I is true and Statement II is true. Statement II is not the correct explanation for Statement I
C) Statement I is true but Statement II is false
D) Statement I is false but Statement II is true
Correct Answer: A
Solution :
Idea In step 1, put n = 1 the obtained result should be a multiple of 7. In step 2, put n = k, take equal to multiple of 7 with any non-zero constant. In step 3, put n = k +1 in the statement and solve till it becomes a multiple of 7. Let P(m):m7 -m \[\therefore \]\[P(1):1-1=0\]is divisible by 7. and P(2): 128 - 2 = 126 is divisible by 7. and P(K): K7 - K is divisible by 7. \[\Rightarrow \]\[{{m}^{7}}-m\]is divisible by 7. \[\{\because m\in N\And K\in N\}\] \[\therefore \]\[P(K+1):{{(K+1)}^{7}}-(K+1)\] \[\Rightarrow \]\[{{K}^{7}}{{+}^{7}}{{C}_{1}}{{K}^{6}}+...{{+}^{7}}{{C}_{6}}(K+1)-K-1\] \[\Rightarrow \]\[{{K}^{7}}{{+}^{7}}{{C}_{1}}{{K}^{6}}+...+7\] \[\therefore P(K+1)\]is divisible by 7 Now, \[{{(K+1)}^{7}}-(K+1)\] is divisible by 7. \[\Rightarrow \]\[{{(m+1)}^{7}}-(m+1)\]is divisible by 7. \[\Rightarrow \]\[{{(m+1)}^{7}}-{{m}^{7}}-1+({{m}^{7}}-m)\]divisible by 7 \[\Rightarrow \]\[{{(m+1)}^{7}}-{{m}^{7}}-1\]is also divisible by 7. TEST Edge In JEE Main, generally multiple and divisibility related questions are asked. To solve these types of questions follow the principle of mathematical induction.
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