Why does every number of shape ababab is divisible by $13$?

Why does it seems like every number $ababab$, where $a$ and $b$ are integers $[0, 9]$ is divisible by $13$?

Ex: $747474$, $101010$, $777777$, $989898$, etc...

Note that $$[ababab] = a\times 101010 + b \times 10101 = 13 (7770a + 777b)$$

Also noteworthy: $$10101 = 1 + 10^2 + 10^4 \equiv \\ 1 + 3^2 + 3^4 = 1 + 9 + 9^2 \equiv\\ 1 +(-4) + (-4)^2 = 1 - 4 + 16 = 13 \equiv 0$$ where $\equiv$ indicates equivalence modulo $13$.

• Can you provide an intuition on how you got to that decomposition? – Juan Jun 7 '16 at 20:57
• @Juan you can look at my answer for the intuition or simply multiply and adding everything in the second line of this answer will tell you the intuition. – Kushal Bhuyan Jun 8 '16 at 2:23
• @Juan I explain how it arises in my answer. – Bill Dubuque Jun 20 '16 at 19:53
• I guess you want $3^2=9$, not $3^2=9^2.$ – user940 Jun 20 '16 at 19:55

These numbers are of the form $(10a+b)\cdot 10101$, and $10101=13\cdot 777$.

• So they are also all divisible by 7 and 37! – Ben Blum-Smith Jun 7 '16 at 15:52
• @BenBlum-Smith Indeed, and by 3. – Christian Sievers Jun 7 '16 at 15:55

Note: $$ab=a\times 10+b\times 1$$

so $$ab\times 100=ab00$$ therefore

$$abab=ab00+ab=ab(100+1)=ab\times 101$$ $$abab00=ab\times101\times100=ab\times10100$$ $$ababab=ab\times10101=ab\times3\times7\times13\times37$$

Hence it is divisible by $3$,$7,13$ and $37$.

A number $ABCDEF$ is divisible by $13$ if and only if $ABC-DEF$ is divisible by $13$.

Note that $\small{ABA-BAB=100(A-B)+10(B-A)+1(A-B)=91A-91B=13(7A-7B)}$.

Therefore $ABA-BAB$ is divisible by $13$.

Therefore $ABABAB$ is divisible by $13$.

• "ABC−DEF is divisible by 13". (Hint of) proof? Or link to? – Rolazaro Azeveires Jun 7 '16 at 19:48

And not only that: Each such number is also divisible by the other primes 3, 7 and 37. Just factor the number 10101! Your specimen number is any 2-digit number $\times 10101\,$.

$abcabc=abc(10^3+10^6)=abc\cdot 1001000=abc\cdot13\cdot77000$.

REMARK.-It is easy generalisable to $abcabcabcabcabc.....abcabc$ for $2n$ times abc.

$ababab$ is a multiple of $10101$, which in turn is a multiple of $13$.

Hint $\,\ {\rm mod}\ 13\!:\,\ 10\equiv 6^2\,\Rightarrow\, 10^6\equiv 6^{12}\equiv 1\,$ by little Fermat.

Hence $\, 0\equiv 10^6-1 \equiv (10^2\!-1)(10^4\!+10^2\!+1) \equiv 99\cdot 10101$

Thus $\,99\not\equiv 0\,\Rightarrow\,10101\equiv 0\,\Rightarrow\, ab\cdot 10101 = ababab\equiv 0,\,$ for $\, 0 \le ab \le 99$

$100^0 \equiv 1 \bmod 13$

$100^1 \equiv 9 \bmod 13$

$100^2 \equiv 3 \bmod 13$

$(ababab)_{10}=c100^2+c100+c =c(100^2+100+1) \equiv c(3+9+1) \equiv 0 \bmod 13$, where $c=10a+b$.