Product of Missing Digits in a Number

[doggydan42]

Homework Statement

Five consecutive multiples of 8 have a 9-digit product of $49xyz2160$. What is the value of $x\cdot y \cdot z$?

Homework Equations

I am unsure of what equations would be relevant.

The Attempt at a Solution

I tried breaking the number into its parts: $4\cdot 10^8+9\cdot 10^7+x\cdot 10^6+y\cdot 10^5+...+6\cdot 10$; though, I wasn't sure what I could do next.

I also realized that the 9-digit number must be divisible by $8\cdot \frac{n!}{(n-5)!}$ for some natural number n.

How would I go about solving this with mathematical rigor, and what would be the fastest way to solve it?

Thank you in advance.

 

[fresh_42]

Homework Statement

Five consecutive multiples of 8 have a 9-digit product of $49xyz2160$. What is the value of $x\cdot y \cdot z$?

Homework Equations

I am unsure of what equations would be relevant.

The Attempt at a Solution

I tried breaking the number into its parts: $4\cdot 10^8+9\cdot 10^7+x\cdot 10^6+y\cdot 10^5+...+6\cdot 10$; though, I wasn't sure what I could do next.

I also realized that the 9-digit number must be divisible by $8\cdot \frac{n!}{(n-5)!}$ for some natural number n.

How would I go about solving this with mathematical rigor, and what would be the fastest way to solve it?

Thank you in advance.

It should be $8^5 \cdot m$ and $(2\cdot 4\cdot 3 \cdot 5 )\,\vert \,m$ because five consecutive numbers have to contain at least a $2,3,4,5$ as divisors. So we search for $(8^6\cdot 15) \,\vert \, 49\ldots$. Well, from here a brute force method works.

Edit: As $2,3,4,5,6$ and $8,9,10,11,12$ can be ruled out quickly, you also know, that $7\,\vert \,m$, so $8^6\cdot 105\,\vert \,49\ldots$ which leaves only a few possibilities.

 

[tnich]

It should be $8^5 \cdot m$ and $(2\cdot 4\cdot 3 \cdot 5 )\,\vert \,m$ because five consecutive numbers have to contain at least a $2,3,4,5$ as divisors. So we search for $(8^6\cdot 15) \,\vert \, 49\ldots$. Well, from here a brute force method works.

Homework Statement

Five consecutive multiples of 8 have a 9-digit product of $49xyz2160$. What is the value of $x\cdot y \cdot z$?

Homework Equations

I am unsure of what equations would be relevant.

The Attempt at a Solution

I tried breaking the number into its parts: $4\cdot 10^8+9\cdot 10^7+x\cdot 10^6+y\cdot 10^5+...+6\cdot 10$; though, I wasn't sure what I could do next.

I also realized that the 9-digit number must be divisible by $8\cdot \frac{n!}{(n-5)!}$ for some natural number n.

How would I go about solving this with mathematical rigor, and what would be the fastest way to solve it?

Thank you in advance.

Call the middle number $k$. Then the $49xyz2160 = 8^5(k-2)(k-1)k(k+1)(k+2)$. I think you could multiple out the polynomial on the right and come up with a very simple and highly accurate approximation by dropping all except one term. Then you could solve for an approximate value of $k$.