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Grade: 11
        Prove 11^n+2+12^2n+1 is divisible by 133 using binomial theorem
one year ago

Answers : (1)

Arun
23044 Points
							
Dear student
 
PART 1: 
First, prove it for the base case of n = 1: 

11^(1+2) + 12^(2+1) 
= 11^3 + 12^3 
= 1331 + 1728 
= 133(10) + 1 + 1728 
= 133(10) + 1729 
= 133(10) + 133(13) 
= 133(26) 

PART 2: 
Assume it is true for a natural number k. Prove it is true for the number k+1: 

True for k: 
11^(k+2) + 12^(2k+1) = 133(m), where m is an integer. 

For k+1: 
11^(k+1+2) + 12^(2(k+1)+1) 
= 11^(k+2) * 11 + 12^(2k + 2 + 1) 
= 11^(k+2) * 11 + 12^(2k + 1) * 12² 
= 11 * 11^(k+2) + (11 + 133) * 12^(2k+1) 
= 11( 11^(k+2) + 12^(2k+1) ) + 133 * 12^(2k+1) 
= 11( 133m ) + 133 * 12^(2k+1) 
= 133 ( 11m + 12^(2k + 1)) 

The stuff in the parentheses will be an integer, so the result is a multiple of 133. 

Therefore by induction a number of that form is divisible by 133. 
 
 
 
Regards
Arun (askIITians forum expert)
one year ago
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