$2^{28} - 1$ is exactly divisible by two numbers between $120$ and $130$ . The sum of these two numbers is

225

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256

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257

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251

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Solution

The correct option is B 256
$2^{28} - 1 = (2^{14})^{2} - (1)^{2}$

$∵ a^{2} - b^{2} = (a + b) (a - b)$

$= (2^{14} + 1) (2^{14} - 1)$

$= (2^{14} + 1) (2^{7} + 1) (2^{7} - 1)$

$= (2^{14} + 1) (127) (129)$

So, $127 a n d 129$ are the two numbers that exactly divide $2^{28} - 1$
and,
$T h e i r s u m i s 256.$

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Similar questions

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Q. Write (T) for true and (F) for false against each of the following statements:
(i) If a number is divisible by 4, it must be divisible by 8.
(ii) If a number is divisible by 8, it must be divisible by 4.
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Question 1

Which of the following statements are true:
(a) If a number is divisible by 3, it must be divisible by 9.
(b) If a number is divisible by 9, it must be divisible by 3.
(c) If a number is divisible by 18, it must be divisible by both 3 and 6.
(d) If a number is divisible by 9 and 10 both, then it must be divisible by 90.
(e) If two numbers are co-primes, at least one of them must be prime.
(f) All numbers which are divisible by 4 must also be divisible by 8.
(g) All numbers which are divisible by 8 must also be divisible by 4.
(h) If a number exactly divides two numbers separately, it must exactly divide their sum.
(i) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately.

Which of the following statements are true?

(a) If a number is divisible by 3, it must be divisible by 9.

(b) If a number is divisible by 9, it must be divisible by 3.

(d) If a number is divisible by 9 and 10 both, then it must be divisible by 90.

(e) If two numbers are co-primes, at least one of them must be prime.

(f) All numbers which are divisible by 4 must also be divisible by 8.

(g) All numbers which are divisible by 8 must also be divisible by 4.

(h) If a number exactly divides two numbers separately, it must exactly divide their sum.

(i) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately.

Q. Which of the following statements are true?

(i) If a number is divisible by 3, it must be divisible by 9.
(ii) If a number is divisible by 9, it must be divisible by 3.
(iii) If a number is divisible by 4, it must be divisible by 8.
(iv) If a number is divisible by 8, it must be divisible by 4.
(v) A number is divisible by 18, if it is divisible by both 3 and 6.
(vi) If a number is divisible by both 9 and 10, it must be divisible by 90.
(vii) If a number exactly divides the sum of two numbers, it must exactly divide the numbers separately.
(viii) If a number divides three numbers exactly, it must divide their sum exactly.
(ix) If two numbers are co-prime, at least one of them must be a prime number
(x) The sum of two consecutive odd numbers is always divisible by 4.

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228−1 is exactly divisible by two numbers between 120 and 130. The sum of these two numbers is

The correct option is B 256228–1=(214)2−(1)2∵a2−b2=(a+b)(a−b)=(214+1)(214–1)=(214+1)(27+1)(27–1)=(214+1)(127)(129)So, 127 and 129 are the two numbers that exactly divide 228−1 and,Their sum is 256.

$2^{28} - 1$ is exactly divisible by two numbers between $120$ and $130$ . The sum of these two numbers is