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Sum of factors

Expert replies
by mathewmithun » Tue Mar 06, 2012 12:19 am
The number 2^48 - 1 divisible by two numbers between 60 and 70. Their sum is:
(a) 124 (b) 128 (c) 132 (d) 136 (e) None of the above

OA: 128
My doubt: how to distinguish between 128 and 132?

My method:
2^48 will have its one's digit as 6 and so the one's place digit of 2^48 - 1 is 5. One of the number between 60 and 70 that can satisfy this condition is 65. the other number should also be between 60 and 70(given condition). So 60<(options-65)<70 is the condition to satisfy. 128 and 132 satisfies (other number will be 63 or 67). Now how to choose between 63 and 67 without the factorization method??? (by factorization method the factors lying between 60 and 70 is 63 and 65)
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Source: — Problem Solving |

by killer1387 » Tue Mar 06, 2012 12:31 am
mathewmithun wrote:The number 2^48 - 1 divisible by two numbers between 60 and 70. Their sum is:
(a) 124 (b) 128 (c) 132 (d) 136 (e) None of the above
use the property
x^n-y^n will always be divisible by x-y
x^n-y^n will be divisible by x+y if n is even

hence 2^48-1=64^8-1
i.e. it is divisible by 64+1=65 and 64-1=63 between 60&70

63+65=128

HTH
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by GMATGuruNY » Tue Mar 06, 2012 3:53 am
mathewmithun wrote:The number 2^48 - 1 divisible by two numbers between 60 and 70. Their sum is:
(a) 124 (b) 128 (c) 132 (d) 136 (e) None of the above

OA: 128
My doubt: how to distinguish between 128 and 132?

My method:
2^48 will have its one's digit as 6 and so the one's place digit of 2^48 - 1 is 5. One of the number between 60 and 70 that can satisfy this condition is 65. the other number should also be between 60 and 70(given condition). So 60<(options-65)<70 is the condition to satisfy. 128 and 132 satisfies (other number will be 63 or 67). Now how to choose between 63 and 67 without the factorization method??? (by factorization method the factors lying between 60 and 70 is 63 and 65)
Follow the factors in red:
2^48 - 1 = (2^24 + 1)(2^24 - 1)
2^24 - 1 = (2^12 + 1)(2^12 - 1)
2^12 - 1 = (2^6 + 1)(2^6 - 1) = 65*63.
65+63 = 128.

The correct answer is B.
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