Let $n$ have $k+1$ decimal digits. If $n$ checks a) and b), obviously $k > 0$. So you're reduced to solve the equation $4\cdot(10u + 6) = u + 6 \cdot 10^k$ in positive integers, making $u$ as small as possible for a $k$-digit decimal number. You need $13v = 10^k - 4$, letting $v = u/2$ be an integer. You check $\mbox{ord}_{13}(10) = 6$ and $10^5 \equiv 4 \bmod 13$. Therefore $13 \mid (10^k - 4)$ iff $k = 6h + 5$, for any $h\in\mathbb{N}$. Then $u = 2 \cdot \frac{10^{6h+5}-4}{13}$, and you're just involved for finding out the least $h\in\mathbb{N}$ such that $u$ has exactly $6h+5$ digits. Thanks to God, $h = 0$ works and $n = 153846$.

This is an alphametric! Set it up like an old-school paper and pencil multiplication problem. _ _ _ _ _ _ _ _ 6 x4 6 _ _ _ _ _ _ _ _ Where we have added extra blanks in case the number is long. Now, 2 _ _ _ _ _ _ _ 4 6 x 4 6 _ _ _ _ _ _ _ 4 1 2 _ _ _ _ _ _ 8 4 6 x 4 6 _ _ _ _ _ _ 8 4 3 1 2 _ _ _ _ _ 3 8 4 6 x 4 6 _ _ _ _ _ 3 8 4 And eventually you get 153846. Sorry the spacing is wrong. How do I fix it?

Here is an even cooler method, I remember from a USAMTS solution to a similar problem: Let the answer be abcdef.... Consider the repeating decimal 0.abcdef....abcdef....abcdef.......=K Then 4K=0.6abcdef....6abcdef....6abcdef....... Note that 10(4K)-6=K Solving, K=2/13 We find the decimal respresentation of 2/13 is .153846153846.....

my soln is somewat dirty write the no in base 10 and then take mod10,100,1000 both sides and done

Jaswinder wrote: my soln is somewat dirty write the no in base 10 and then take mod10,100,1000 both sides and done Jaswinder, the postings should not be in hints but the complete solution by Zetax (http://www.artofproblemsolving.com/Forum/viewtopic.php?t=135914) Generalistation, a way of analysing it: let the digits of the originally $n$ digit number be $x$ , denoted by $ a_n-1, a_n-2........,a_1,a_0=6$ $x$ =$10^{n-1}a_n-1+ 10^{n-2}*a_n-2+....+10a_1+ 6 $ according to this problem $ 4x=6*10^{n-1}+(10^{n-2}a_n-1+10^{n-3}a_n-2+....+10a_2+a_1 )$ It is equal to this binomial exp: $ 4x=6*10^{n-1}+ \frac{x-6}{10}$ $40x=6*10^n +x-6$ $13x=2(10^{n}-1)$ $ \rightarrow $ this shows that $10^{n}-1$ (a number containing only $9$) must be divisible by $13$ $999999 = 13*76923 \rightarrow x= 2*76923 = 153846 $ this solution is rather guessing out , and it's obvious this solution doesn't fetch us answer to these kind of problems . I just posted it to make sure there was another indirect solution for this problem . Readers don't take this post important as i mentioned u above

getting our hands dirty,we easily get the number $153846$ and indeed is $153846*4=615384$.It is from Imo $1962$ http://www.artofproblemsolving.com/Forum/resources.php?c=1&cid=16&year=1962&sid=7d944a84d144d5ce1f9d027cb681a3ad

Salutations, for uvwxyz in range(100000,250000): z = uvwxyz % 10 uvwxy = uvwxyz // 10 zuvwxy = 100000 * z + uvwxy if zuvwxy == 4 * uvwxyz: print(uvwxyz)for uvwxyz in range(100000,250000): z = uvwxyz % 10 uvwxy = uvwxyz // 10 zuvwxy = 100000 * z + uvwxy if zuvwxy == 4 * uvwxyz: print(uvwxyz)RunResetPop Out /

Let $n=10^k a_k+10^{k-1}a_{k-1}+\dots +10a_1+6$ be the number. Writing $n$ in the form $n=10N+6,$ where $10^{k-1}<N<10^k,$ we have from (b), $$4(10N+6)=6\cdot 10^k+N\implies 13N=2(10^k-4) \implies (-3)^k \equiv 4 \pmod{13}.$$We find $k=5$ as the least that satisfies. So $N=15384$ and $\boxed{n=153846}.$

Let $a$ be the other part. Let $b$ be the number of digits in $a$. $4(10a+6)=6*10^b+a$ $40a+24=6*10^b+a$ $39a+24=6*10^b$ $13a+8=2*10^b$ Testing small values of $b$, get that $b=5$ works and results in $a=15384$. So the answer is $\boxed{153846}$

aight so basically _6 times four is 6_ clearly _6 times four ends with 4 so we want _46 times four to be 6_4 clearly _46 times four ends with 84 so we want _846 times four to be 6_84 clearly _846 times four ends with 384 so we want _3846 times four to be 6_384 clearly _3846 times four ends with 5384 so we want _53846 times four to be 6_5384 clearly _53846 times four ends with 15384 so we want _153846 times four to be 6_15384 oh wait $n=\boxed{153846}$ times four is 615384