Determinate-m-such-that-m-is-written-abcca-in-base-5-and-is-written-bbab-in-base-8-

Question Number 124694 by mathocean1 last updated on 05/Dec/20

D e t e r m i n a t e m s u c h t h a t

m i s w r i t t e n a b c c a i n b a s e 5 a n d i s

w r i t t e n b b a b i n b a s e 8 .

Answered by floor(10²Eta[1]) last updated on 05/Dec/20

M = {abcca}_{5} = {bbab}_{8}; 1 ⩽ a ⩽ 4, 0 ⩽ b, c ⩽ 4

a . 5^{4} + b . 5^{3} + c . 5^{2} + c . 5 + a = b . 8^{3} + b . 8^{2} + a . 8 + b

\Rightarrow 309 a - 226 b + 15 c = 0

309 a - 226 b = - 15 c < 0

309 a < 226 b \Rightarrow a < b

309 a - 226 b + 15 c \equiv 4 a - b \equiv 0 (\mod 5)

4 a \equiv b (\mod 5) \Rightarrow (a, b) = (2, 3), (1, 4)

checking you see that (a, b) = (2, 3) \Rightarrow c = 4

M = 23442_{5} = 3323_{8}