Question

The number of 6-digit numbers of the form ababab (in base 10) each of which is a product of exactly 6 distinct primes is

• A. 8
• B. 10
• C. 13
• D. 15

Answer 1

The correct option is C

13

$N=ababab$
$1<a\le 9,0<b\le 9;a,b\in I$
$N={10}^{5}a+{10}^{4}b+{10}^{3}a+{10}^{2}b+10a+b$
$=({10}^4+{10}^2+1)(10a+b)$
$=({10}^2+10+1)({10}^2-10+1)(10a+b)$
$=3\times 37\times 13\times 7(10a+b)...\left(1\right)$
then $10a+b=P1\times P2$
$P1,P2\in$ prime and $10\le 10a+b\le 99$

$\begin{array}{ccc}a& b& 10a+b\\ 1& 0& 10=2\times 5\\ 2& 2& 22=2\times 11\\ 3& 4& 34=2\times 17\\ 3& 8& 38=2\times 19\\ 4& 6& 46=2\times 23\\ 5& 5& 55=5\times 11\\ 5& 8& 58=2\times 29\\ 6& 2& 62=2\times 31\\ 7& 4& 74=2\times 37\\ 8& 2& 82=2\times 41\\ 8& 5& 85=5\times 17\\ 9& 4& 94=2\times 47\\ 9& 5& 95=5\times 19\end{array}$