Answer:
The required integer is 157.
Step-by-step explanation:
Let the unknown number be x.
It is given that the remainder 1 when x is divided by 13, and remainder 2 when x is divided by 31.
[tex]x\equiv 1(mod 13)[/tex]
[tex]x\equiv 2(mod 31)[/tex]
Here, [tex]a_1=1,a_2=2,n_1=13,n_2=31[/tex]
13 and 31 hare prime numbers. so the GCD o 13 and 31 is
[tex]N=GCD(13,31)=13\times 31=403[/tex]
[tex]N_1=\frac{N}{13}=31[/tex]
[tex]N_2=\frac{N}{31}=13[/tex]
Using Chinese remainder theorem, we get
[tex]y_1=31^{-1}(mod 13)=8[/tex]
[tex]y_1=13^{-1}(mod 31)=12[/tex]
The formula to find the value of x is
[tex]x\equiv (a_1y_1N_1+a_2y_2N_2)(mod N)[/tex]
Substitute the given values in the above formula.
[tex]x\equiv (1\cdot 8\cdot 31+2\cdot 12\cdot 13)(mod 403)[/tex]
[tex]x\equiv 560(mod 403)[/tex]
[tex]x=560+403n[/tex]
Let n=-1, because we need to find the least positive integer.
[tex]x=560+403(-1)[/tex]
[tex]x=560-403[/tex]
[tex]x=157[/tex]
Therefore, the required integer is 157.
