There are some basic formulas in the RSA Algorithm:
public keys are: e, n
private keys are: d,n
let us consider two relatively large co-prime numbers p and q then
1. $n = p*q$
2. $\phi(n)= (p-1)*(q-1)$
3. $ed= 1 mod \phi (n)$
In the question they have mentioned that:
$p=13, q=17 ,e=35$
So $\phi(n)= (13-1)(17-1)=12*16=192$
$ed=1 mod \phi(n)$
$35*d=1 mod 192$; here the tricky part comes in the role,
In 192 the last digit is 2 and in 35 the last digit is 5. Now d's last digit must be that digit which after multiplying with 5 gives 1 as the remainder when the mod 2 is applied on the product.
So the last digit of 'd' contains 1, 3, 5, 7, 9 and the most important thing both e and d are always co-prime. Now you can take some values and check the condition.
11 is the smallest value of 'd' satisfying all the properties.
Answer : d=11