Q1-x-lt-1a-1-a-2-a-n-gt-N-amp-y-lt-a-1-a-2-a-n-1-gt-N-if-y-3x-then-find-the-smallest-value-of-x-Q2-with-the-above-conditions-what-other-values-can-be-placed-besides-the-n

Question Number 192172 by mehdee42 last updated on 10/May/23

Q 1 ∴ x =< 1 a_{1} a_{2} \dots a_{n} >\in N & y =< a_{1} a_{2} \dots a_{n} 1 >\in N

i f y = 3 x t h e n, f i n d t h e s m a l l e s t

v a l u e o f x

Q 2 ∴ w i t h t h e a b o v e c o n d i t i o n s, w h a t o t h e r v a l u e s

c a n b e p l a c e d b e s i d e s t h e n u m b e r “ 1^{″}

Answered by AST last updated on 10/May/23

1) x = 10^{n} + \frac{y - 1}{10}; y = 3 x \Rightarrow x = 10^{n} + \frac{3 x - 1}{10}

\Rightarrow 10 x = 10^{n + 1} + 3 x - 1 \Rightarrow 7 x = 10^{n + 1} - 1

10^{n + 1} - 1 \equiv 0 (m o d 7) \Rightarrow 3^{n + 1} \equiv 1 (m o d 7) \Rightarrow 3^{n} \overset{7}{\equiv} 5

{o r d}_{3} (7) = 6; 3^{5} \overset{7}{\equiv} 5; ϕ (7) = 6 \Rightarrow n = 6 k + 5; m i n (n) = 5

\Rightarrow m i n (x) = \frac{10^{6} - 1}{7} = 142857

2) x = p 10^{n} + \frac{3 x - p}{10} \Rightarrow 10 x = p 10^{n + 1} + 3 x - p

\Rightarrow 7 x = p (10^{n + 1} - 1)

T h e f i r s t d i g i t o f \frac{p (10^{n + 1} - 1)}{7} s h o u l d b e p

T h i s i s o n l y t r u e f o r p = 1, 2

\Rightarrow p = 1 a n d 2 a l w a y s w o r k w h e n n = 6 k + 5 .

H e n c e, o n l y 2 c a n b e p l a c e d b e s i d e s t h e n u m b e r

“ 1^{″} .

Commented by mehdee42 last updated on 10/May/23

v e r y n i c e S i r

f o r p = 2; t h e l o w e s t v a l u e

x = 285714