Track / Stage 4 / 167 of 340 #427 of 2000

Problem 427

AMC 12 late, AIME early

Number theory Difficulty 4.8 Find the answer

6. Find the smallest natural number ending in the digit 2 that doubles when this digit is moved to the beginning.

Your answer

Official solution

Answer: 105263157894736842

Solution: Let's write the number in the form *... 2 and gradually restore the "asterisks" by multiplying by 2:

$* * * . . . * * 2 \times 2=* * * . . * * 4$

$* * * . . . * 42 \times 2=* * * . . . * 84$

$* * * \ldots * 842 \times 2=* * * \ldots * 684$

$* * * . . * 6842 \times 2=* * * \ldots * 3684$

***...*36842 $2=* * * \ldots * 73684$

***... $* 736842 \times 2=* * * . . . * 473684$

***...*4736842 x $2={ }^{* * *} \ldots . . * 9473684$

...

$105263157894736842 \times 2=2105263157$

89473684

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.