@iLoveTorres said:A no N when increased by 50% has its no of factors as unchanged . But when decreased by 25% , the no of factors is decreased by 33.33%. Min value of N ?
let the number be
N = (2^a)*(3^b)*(5^c)....................
no. of factors = (1+a)(1+b)(1+c).......
to increase by 50% then we are looking for factors of 3*N/2
hence the number of factors = (a+1 - 1)}*(b+1 +1)*(c+1)..............
since the number of factors remain the same hence
(1+a)(1+b)(1+c)............ = a(b+2)(1+c)...........
=>(1+a)(1+b) = a(b+2)
=>(1 + a + b + ab) = ab + 2a
=> 1 + b = a ..................(i)
to decrease by 25% we multiply the number by 3/4 hence the number of factors become
(1+a-2)(1+b+1)(1+c)...........
since the number of factors decrease by 33.33% hence
(a-1)(b+2)(1+c).............. = (3/2)*(1+a)(1+b)(1+c).............
=>3*(a-1)(b+2) = 2*(1+a)(1+b)....................(ii)
=> 3*(a-1)(a+1) = 2*(1+a)a [replacing 1+b = a]
=>3(a^2 - 1) = 2a + 2a^2
=>a^2 - 2a - 3 = 0
=>(a - 3)(a - 1) = 0
=>a =3 or a=1 clearly a =1 is not possible as in that case the number of factors become zero when decreased by 25%
if a = 3 =>b=2
hence all the numbers of form
(2^3)*(3^2)*5^c........... will satisfy the condition.
This is essentially same as what
@scrabbler bhai has done earlier in one of this post.
ATDH.
