# 100 followers' problem (number theory)

Number Theory Level 5

Consider the decimal expansion of

$$\frac{10^{n}}{x}$$

where $$n,x$$ are integers.

Let $$\frac{10^{n}}{x}= a_{1}a_{2}...a_{n} . a_{n+1}a_{n+2}.......$$ where the single dot represents the decimal point!!

given that

$$\sum_{i=1}^{6p} a_{i} = 2p$$

$$\sum_{i=1}^{6q} (-1)^{i+1}a_{i} = 0$$

find $$x$$

×