The sum of all the elements of the set $\{\alpha \in \{1,2,\mathrm{.....},100\}:\text{HCF}(\alpha ,24)=1\}$  is ___.

 $\text{HCF}(\alpha ,24)=1$
Now,  $24=2^2.3$
$\to \alpha$  is not the multiple of 2 or 3
Sum of values of  $\alpha$
$=S\left(U\right)-\{S$ (multiple of 2) + S (multiple of 3) - S(multiple of 6) $\}$
  $(1+2+3+\mathrm{......100})-(2+4+\mathrm{6.....}+100)-(3+6+\mathrm{.....99})+(6+12+\mathrm{....}+96)$

$=\frac{100\times 101}{2}-50\times 51-\frac{33}{2}\times (3+99)+\frac{16}{2}(6+96)$
 $=5050-2550-1683+816=1633$