Question

If n is a positive integer,let s(n) denote the integer obtained by removing the last digit of n and placing it in front.for example,s(731)=173.what is the smallest positive integer n ending in6 satisfying s(n)=4n?

Answer 1

Let n=10a+6, and a has d digits.

S(n)=4n⟹6×10^d+a=4n=4(10a+6)

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6×10^d−24=39a

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2×10^d−8=13a

2×10^d≡8(mod13)

10^d≡4(mod13),(2,13)=1

10^5≡4(mod13)

and 5 is the minimum number to fulfill this.

a=2×10^5−813=15384

and n=15384×10+6=153846