What is the maximum value of $n$ such that $7 \times 343 \times 385 \times 1000 \times 2401 \times 77777$ is…
What is the maximum value of $n$ such that $7 \times 343 \times 385 \times 1000 \times 2401 \times 77777$ is divisible by $35^n$?
3
4
5
7
Solution
$35^n = 5^n \times 7^n$. Count powers of 5 and 7. Powers of 7: $7$ gives $7^1$, $343=7^3$, $385=5\times7\times11$ gives $7^1$, $1000$ none, $2401=7^4$, $77777=7\times11111$ gives $7^1$ - total $7$ power $=1+3+1+0+4+1=10$. Powers of 5: $385$ gives $5^1$, $1000=2^3\times5^3$ gives $5^3$ - total $5$ power $=4$. The limiting factor is the power of 5, which is 4. Hence maximum $n=4$.