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

Answer: (C)

$13$

$ababab$ in decimal system is ${10}^{5}a+{10}^{4}b+{10}^{3}a+{10}^{2}b+10a+b=10101(10a+b)=10101\times (ab{)}_{10}$

It is known that $ababab$ is a product of $6$ distinct primes.

Also, $10101=3\times 7\times 13\times 37$

Thus, $(ab{)}_{10}$ has to be expressed as a product of $2$ prime numbers, none of them from $\{3,7,13,37\}$

Such possible prime pairs are $(2,5),(2,11),(2,17),(2,19),(2,23),(2,29),(2,31),(2,41),(2,43),(2,47),(5,11),(5,17),(5,19)$

A total of 13 such numbers are possible.