Let N=24500, then find :-(i) The number of ways by which N can be resolved into two factors.(ii) The number of ways by which 5N can be resolved into two factors.

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Let $N = 24500, then find$ :-

(i) The number of ways by which N can be resolved into two factors.

(ii) The number of ways by which 5N can be resolved into two factors.

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(i) N = 23; (ii) 5 N = 56

(i) N = 18; (ii) 5 N = 56

(i) N = 18; (ii) 5 N = 23

None of these

answer is C.

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Detailed Solution

Concept- We will first find the factors of the given number, then we will compare the factors with

N = x^{a} y^{b} z^{c}

, we will get the value of

a, b, c

then we will substitute the values of

a, b, c

in,

N = \frac{1}{2} (a + 1) (b + 1) (c + 1)

, formula to get the number of ways.
(i) Factors of

24500 = 2^{2} {\times 5}^{3} {\times 7}^{2}

On comparing this

N = x^{a} y^{b} z^{c}

, we get,

a = 2, b = 3, c = 2,

N = 24500

Using formula:-

N = \frac{1}{2} (a + 1) (b + 1) (c + 1)

N = \frac{1}{2} (2 + 1) (3 + 1) (2 + 1)

N = 18

(ii) Given

5 N,

which becomes

5 \times 24500

5 N = 5 \times 24500

Factors of

5 \times 24500 = 2^{2} {\times 5}^{4} {\times 7}^{2}

On comparing this

N = x^{a} y^{b} z^{c}

, we get,

a = 2, b = 4, c = 2,

N = 5 \times 24500

Using formula:-

N = \frac{1}{2} [(a + 1) (b + 1) (c + 1) + 1]

N = \frac{1}{2} [(2 + 1) (4 + 1) (2 + 1) + 1]

N = 23

Hence, the correct answer is option 3.

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