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Funny Factorisation

Age 11 to 16 Challenge Level:

Some 4 digit numbers can be written as the product of a 3 digit number and a 2 digit number using each of the digits $1$ to $9$ once, and only once.

The number $4396$ can be written as just such a product.
calculation showing a 3-digit number multiplied by a 2-digit number, giving the answer 4396

Can you find the factors?

Maths is full of surprises!
The numbers $5796$ and $5346$ can each be written as a product like this in two different ways.

Four calculations showing 3-digit numbers multiplied by 2-digit numbers. Two have the answer 5796 and the other two have the answer 5346.

Can you find these four funny factorisations?
 

Extension

There are two more funny factorisations to find, using each of the digits $1$ to $9$ once, and only once.
Can you fill in the blanks in the multiplication below to find one of them?
A 3-digit number with units digit 9, multiplied by a 2-digit number with tens digit 4, giving a 4-digit product with hundreds digit 6

If you know a bit about computer programming, you may wish to write a program to find the final funny factorisation.