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I have a problem that gives me this data: $$(e,n) = (13,1563151)$$ One of the primes $(p,q)$ is $<1050$.

With this data I shall calculate the private key $d$.

I can only use a simple arithmetic calculator for this problem.

I figured out that $d=e^{-1} \bmod \phi(N)$, $\phi(N)=(p-1)(q-1)$ so I should somehow get $p$ and $q$ with the data I'm given but I'm stuck there.

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I don't see better than intelligent brute force. Start down from $1050$ looking for primes. You know that primes end in $1,3,7,9$ and are not divisible by $3$, so try $1049,1043, 1039, 1037, 1033, 1031, 1027, 1021$ and bingo.

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    I've thought of this, but seems rather unlikely, given that it is an exam question and we do not have much material to work with. Weird. – 2017-01-25
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    I don't think it is unlikely if you are allowed calculators. Especially if you have a memory available so you don't punch the seven digit number in each time, this will be quick. The student who knows the facts I gave will only try eight times compared to one who has to try $29$ due to not sorting out the obvious non-primes. Of course, you still have to find $d$ – 2017-01-25