If n is the number of irrational terms in the expansion of 3 1 / 4 + 5 1 / 8 60 , then n - 1 is divisible by :

${\left(3^{1/4}+5^{1/8}\right)}^{60}$

$={}^{60}C_r{\left(3^{1/4}\right)}^{60-r}.{\left(5^{1/8}\right)}^r$

$={}^{60}C_r{\left(3\right)}^{\frac{60-r}{4}}.5^{\frac{r}{8}}$

For rational terms.

$\frac{r}{8}=k; 0\le r\le 60$

$0\le 8k\le 60$

$0\le k\le \frac{60}{8}$

$0\le k\le 7.5$

$k=0,1,2,3,4,5,6,7$

$\frac{60-8k}{4}$ is always divisible by $4$ for all value of $k$

Total rational terms $=8$

Total terms $=61$

irrational terms $=53$

$n-1=53-1=52$

$52$ is divisible by $26$.