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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