Question Number 10521 by ajfour last updated on 16/Feb/17

A number (αβ..λ...μ2)×2 =(2αβ..λ...μ)  find the number.

Answered by mrW1 last updated on 17/Feb/17

(Part I)  let x=(αβ..λ...μ)=a number with n−1 digits  let y=(αβ..λ...μ2)=a number with n digits  let z=(2αβ..λ...μ)=a number with n digits  y=x×10+2=10x+2  z=2×10...0+x=2×10^(n−1) +x    (αβ..λ...μ2)×2 =(2αβ..λ...μ)  ⇔y×2=z  (10x+2)×2=2×10^(n−1) +x  19x=2×10^(n−1) −4=2(10^(n−1) −2)  x=((2(10^(n−1) −2))/(19))    the first suitable number is n=18 and  x=((2(10^(17) −2))/(19))=10526315789473684  y=105263157894736842  (105263157894736842)×2=(210526315789473684)    the second suitable number is n=36 and  x=((2(10^(35) −2))/(19))=10526315789473684210526315789473684  y=105263157894736842105263157894736842  (105263157894736842105263157894736842)×2=(210526315789473684210526315789473684)    the third suitable number is n=54 and  x=((2(10^(53) −2))/(19))=10526315789473684210526315789473684210526315789474684  y=105263157894736842105263157894736842105263157894736842  (105263157894736842105263157894736842105263157894736842)×2=(210526315789473684210526315789473684210526315789473684)    the general solution is n=18i with i=1,2,...  and the number is  y=105263157894736842105263157894736842105263157894736842105263157894736842......

Commented byajfour last updated on 16/Feb/17

great explanation, i cud arrive at the answer but ur explanation is elaborate. but wait, it should end with.... 736842.typing error probably.

Commented bymrW1 last updated on 16/Feb/17

you are right. i had a typing error.   now it′s fixed. thank you for pointing out that.