is this known and what is it called?

Question

I noticed that XXX...n * Y = YYY...n * X

for example

999 * 8 = 888 * 9 = 7992

77777 * 9 = 99999 * 7 = 699993

13 * 777 = 7 * (1300 + 130 + 13 ) = 10101

19 * 7777 = 7 * ( 19000 + 1900 + 190 + 19) = 147763

what is this property called? or is it known before?

Answer 1

This property is called "representation in base 10" and "the multiplication is commutative". For any $X$ and $Y$ the following equalities hold: $$ \underbrace{XX\ldots X}_{n\text{ times}}=X\cdot \underbrace{11\ldots 1}_{n\text{ times}} $$ So $$ \underbrace{XX\ldots X}_{n\text{ times}}\cdot Y=X\cdot Y\cdot \underbrace{11\ldots 1}_{n\text{ times}}=\underbrace{YY\ldots Y}_{n\text{ times}}\cdot X $$