I have the following quiz but I don't really know how to solve it.
Find the smallest positive integer that starts (in decimals) with $2016$ and is divisible by $2017$.
Why to they specify the decimals in brackets?
First thought was to divide $2016$ by $2017$. $\dfrac{2016}{2017} = 0.999504$.
Should the integer start with $2016$? I find it confusing. Can someone help?
Best Answer
Your first thought is actually quite spot-on.
Since $2016/2017 \approx 0.999504$, we have $$0.9995 \times 2017 < 2016 < 1\times 2017$$
Multiplying this by various powers of $10$ yields
$$9\times 2017 < 10 \times 2016 < 10 \times 2017$$ $$99\times 2017 < 100 \times 2016 < 100 \times 2017$$ $$999\times 2017 < 1000 \times 2016 < 1000 \times 2017$$
The above shows that there are no $5,6,7$-digit multiple of $2017$ starting with $2016$. Finally:
$$9995\times 2017 < 10000 \times 2016< 9996 \times 2017 = 20161932$$