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

256

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248

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251

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157

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Solution

The correct option is A 256
$2^{28} - 1 = (2^{7})^{4} - (1)^{4}$
So, it must be divisible by $(2^{7} + 1)$ i.e. 129 and $(2^{7} - 1)$
i.e. 127
So, the answer is 129 + 127 = 256

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Q. How many numbers between 1 and 200 are exactly divisible by exactly two of 3, 9 and 27?

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.
(iii) If a number divides the sum of two numbers exactly, it must exactly divide the numbers separately.
(iv) If a number is divisible by both 9 and 10, it must be divisible by 90.
(v) A number is divisible by 18 if it is divisible by both 3 and 6.
(vi) If a number is divisible by 3 and 7, it must be divisible by 21.
(vii) The sum of two consecutive odd numbers is always divisible by 4.
(viii) If a number divides two numbers exactly, it must divide their sum exactly.

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.

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 in between 120 and 130. What is the sum of these two numbers?

The correct option is A 256228−1=(27)4−(1)4 So, it must be divisible by (27+1) i.e. 129 and (27−1) i.e. 127 So, the answer is 129 + 127 = 256

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

The correct option is A 256
$2^{28} - 1 = (2^{7})^{4} - (1)^{4}$
So, it must be divisible by $(2^{7} + 1)$ i.e. 129 and $(2^{7} - 1)$
i.e. 127
So, the answer is 129 + 127 = 256