Two numbers that multiply to a product that contains the original digits

Question

Recently I found an interesting combination of factors that forms a product that contains the original digits from those factors, as presented below:

$$86 * 8 = 688.$$

Is there a name for these types of factors and products or is this just a coincidence?

You might like to look at Vampire numbers, including pseudovampire numbers as a variant. More generally, when basic arithmetic operators are allowed, these are known as Friedman numbers. — nickgard, Jun 11 '17 at 09:24

score 1 · Answer 1 · answered Jun 04 '19 at 14:05

$$86*8=688$$

It is not quite a coincidence.

You can write $$86 = 8 * 10^1 + 6*10^0$$

Using this form, Express: $$86*8=688$$

$$(8 * 10^1 + 6*10^0)*8=6*10^2 + 8*10^1+8*10^0$$

To make this form general, we can write the above as:

$$(x * 10^1 + y*10^0)*x=y* 10 ^2 + x*10^1+x *10^0$$

Simplify to get:

$$(10x+y)x=100y+11x$$

$$10x^2+yx-100y-11x=0$$

The above equation has many solutions, but for $x,y$ less than $10$ and positive, the only solution is:

$$(x,y):(8,6)$$

The point of all this is that the equality is not a "coincidence".

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