Q. 45.0( 2 Votes )

If 49A and A49, where A > 0, have a common factor, find all possible values of A.

Answer :

Given that, A > 0.

We need to find the possible values of A, such that 49A and A49 have a common factor.


A can take values from 1, 2, 3, 4, 5, 6, 7, 8, 9.


If A = 1,


We have 491 and 149.


Split 491, we get 491 = 491 × 1 [ 491 is a prime number]


Split 149, we get 149 = 149 × 1 [ 149 is a prime number]


So, 491 and 149 have no common factors.


If A = 2,


We have 492 and 249.


Split 492, we get 492 = 2 × 2 × 3 × 41


Split 249, we get 249 = 3 × 83


So, 492 and 249 have a common factor, 3.


If A = 3,


We have 493 and 349.


Split 493, we get 493 = 17 × 29


Spit 349, we get 349 = 349 × 1 [ 349 is a prime number]


So, 493 and 349 have no common factors.


If A = 4,


We have 494 and 449.


Split 494, we get 494 = 2 × 13 × 19


Split 449, we get 449 = 449 × 1 [ 449 is a prime number]


So, 494 and 449 have no common factors.


If A = 5,


We have 495 and 549.


Split 495, we get 495 = 3 × 3 × 5 × 11


Split 549, we get 549 = 3 × 3 × 61


So, 495 and 549 have 2 common factors, 3 and 3.


If A = 6,


We have 496 and 649.


Split 496, we get 496 = 2 × 2 × 2 × 2 × 31


Split 649, we get 649 = 11 × 59


So, 496 and 649 have no common factors.


If A = 7,


We have 497 and 749.


Split 497, we get 497 = 7 × 71


Split 749, we get 749 = 7 × 107


So, 497 and 749 has a common factor, 7.


If A = 8,


We have 498 and 849.


Split 498, we get 498 = 2 × 3 × 83


Split 849, we get 849 = 3 × 283


So, 498 and 849 has a common factor, 3.


If A = 9,


We have 499 and 949.


Split 499, we get 499 = 499 × 1 [ 499 is a prime number]


Split 949, we get 949 = 13 × 73


So, 499 and 949 have no common factors.


Hence, A = 2, 5, 7 and 8.


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