Question

Srikanth has made a project on real numbers, where he finely explained the applicability of exponential laws and divisibility conditions on real numbers. He also included some assessment questions at the end of his project as listed below. Answer them.
(i) For what value of n, 4n ends in 0?
(a) 10 (b) when n is even
(c) when n is odd (d) no value of n(ii) If a is a positive rational number and n is a positive integer greater than 1, then for what value of n, an is a rational number?
(a) when n is any even integer (b) when n is any odd integer
(c) for all n > 1 (d) only when n = 0(iii) If x and yare two odd positive integers, then which of the following is true?
(a) x2 + y2 is even (b) x2 + y2 is not divisible by 4
(c) x2 + y2 is odd (d) both (a) and (b)(iv) The statement 'One of every three consecutive positive integers is divisible by 3' is
(a) always true (b) always false
(c) sometimes true (d) None of these(v) If n is any odd integer, then n2 - 1 is divisible by
(a) 22 (b) 55 (c) 88 (d) 8

Answer 1

Answer:

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Answer 2

Answer:

(i) Option (1) 10 is the correct answer.

(ii) Option (c) for all n > 1 is the correct answer.

(iii) Option (a) is True.

(iv) Option (a) always true is the correct answer.

(v) Option (d) 8 is the correct answer.

Step-by-step explanation:

(i) For what value of n, 4n ends in 0?

Since n is a real number and it is multiplied by 4 = $2^{2}$. For the product to end in zero or have zero in its units place n should be either 5 or end with 5 or is a multiple of 2 and 5.

10 is a multiple of 2 and 5.

Option (1) 10 is the correct answer.

(ii) a is a positive rational number and n is a positive integer greater than 1, then for what value of n, an is a rational number?

From the fundamentals of rational numbers, the product of two rational numbers is always a rational number.

• An integer is also a rational number. So n is also a rational number along with a.
• So the product of a and n will always be a rational number.

Option (c) for all n > 1 is the correct answer.

(iii)If x and y are two odd positive integers, then which of the following is true?

Option (a) $x^{2} +y^{2}$ is even

• Product of two odd numbers is always an odd number. So $x^{2}$ and $y^{2}$ are odd numbers.
• Sum of two odd numbers is always an odd number. So the sum of  $x^{2}$ and $y^{2}$ is even.

Option (a) is True.

Option (b) is false as sum of two odd numbers is not odd.

(iv)The statement 'One of every three consecutive positive integers is divisible by 3' is

Option (a) always true is the correct answer.

Explanation:

Consider 3 consecutive positive integers n, (n + 1), (n + 2)

Let r be the remainder when divided by 3, q be the quotient.

n can written in terms of q, r as n = 3q + r

n+1 = 3q+r+1

n+2 = 3q+r+2

0 ≤ r < 3: n = 3q or 3q+1 or 3q+2.

When n = 3q, n+1 and n+2 are not divisible by 3.

When n = 3q+1, n+2 is divisible by 3, n, n+1 are not.

When n = 3q+2, n+1 is divisible by 3, n, n+2 are not.

So, One of every three consecutive positive integers is divisible by 3 is always true.

(v) If n is any odd integer, then $n^{2} -1$ is divisible by

n is odd number, so it is in the form (2x + 1). Substituting in $n^{2} -1$,

$(2x+1)^{2} -1\\= (2x+2)(2x)\\=4(x+1)x$

If x is even number or odd, one term is even and the other is odd number.

So the product x(x+1) is always divisible by 2.

Hence the product 4(x+1)x is always divisible by 8.

If n is any odd integer, then $n^{2} -1$ is divisible by 8.

Option (d) 8 is the correct answer.

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