Question

if sum of 2 smallest factor and 2 largest factors of a number N are 5 and 90 respectively. then find the value of N​

Answer 1

Solution :-

we know that, we can write any number N in the form of product of their prime factors .

→ N = x^a * y^b * z^c

→ Total number of factors of N = (a + 1) * (b + 1) * (c + 1)

Case 1) :- when 1 as a smallest factor included and largest factors N is also included :-

A/q,

→ 1 + x = 5

→ x = 4

Conclusion :- if x = 4, number N is divisible by 2 also . Then, two smallest factors will be 1 and 2 . But there sum is not equal to 5 .

Case 2) :- In smallest factors 1 is not included and in two largest factors N is included .

→ → Sum of 2 smallest factors of N = 5

So,

→ Two smallest factors are = 2 and 3 . { 2 + 3 = 5 } .

then,

→ Sum of two largest factors of N = 90

→ N + (N/2) = 90

→ (3N/2) = 90

→ N = (180/3)

→ N = 60 .

Verification :-

→ 60 = 2² * 3¹ * 5¹

So,

→ Total factors of 60 = (2 + 1) * (1 + 1) * (1 + 1) = 3 * 2 * 2 = 12

then,

→ Factors of 60 are = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.

therefore,

→ Sum of smallest two factors (Other than 1) = 2 + 3 = 5 .

→ Sum of largest two factors = 30 + 60 = 90 .

Hence, value of N is equal to 60 .

Case 3) :- In smallest factors 1 is not included and in two largest factors N is also not included .

→ Sum of 2 smallest factors of N = 5

So,

→ Two smallest factors are = 2 and 3 . { 2 + 3 = 5 } .

then,

→ Sum of two largest factors of N = 90

→ (N/2) + (N/3) = 90

→ (3N + 2N)/6 = 90

→ 5N = 90*6

→ N = 108 .

Hence, the value of N is equal to 108 .

Verification :-

→ 108 = 2² * 3³

→ Total factors of 108 = (2 + 1) * (3 + 1) = 3 * 4 = 12 .

So,

→ Factors of 108 are = 1, 2, 3 , 4, 6, 9, 12, 18, 27, 36, 54 and 108.

then,

→ Sum of smallest two factors (Other than 1) = 2 + 3 = 5 .

→ Sum of largest two factors (Other than 108) = 36 + 54 = 90 .

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Answer 2

Given: Sum of 2 smallest factor and 2 largest factors of a number N are 5 and 90 respectively

To Find : The value of N

Solution:

Smallest factor of  any number is always 1

and largest factor is the number itself

Hence smallest factor = 1

Largest factor = N

Sum of   2 smallest factor  = 5

Hence other smallest factor = 5 - 1  = 4

1 * N  =  4 *  2nd largest factor

=> 2nd largest factor = N/4

N/4  + N  = 90

=> 5N = 360

=> N = 72

But 72 has factors  2 and 36 as well  

so 1 and 4 are not 2 smallest factor.

Hence 72 is not possible    

So there does not exist any such N

Hence there must be some mistake in data

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