So many Factors

Number Theory Level pending

$X = a^{p}b^{q}c^{r}s^{t}$ has $2016$ Factors

$Y = b^{p}d^{q}a^{s}$ has $168$ Factors

$Z = c^{q}b^{r}d^{s}$ has $288$ Factors

Find Minimum possible value of $p+q+r+s$

1 solution

Rishabh Deep Singh
Dec 23, 2017

X has 2016 Factors

$(p+1)(q+1)(r+1)(s+1) = 2016$ ...........(1)

Y has 168 Factors

$(p+1)(q+1)(s+1) = 168$ .................(2)

Z has 288 Factors

$(q+1)(r+1)(s+1) = 288$ ............. (3)

From ${(1)}/{(2)}$ $\quad\quad\quad\quad$ r = 11

From ${(1)}/{(3)}$ $\quad\quad\quad\quad$ p = 6

Substituting Value of p in $(2)$

$(q+1)(s+1) = 24$

Possible cases for $(q,s)$ are $(0,23),(1,11),(2,7),(3,5)$

Minimum Value of $p+q+r+s = 6+3+11+5 = 25$