Finding ? to Make A Multiple of 7, 11, and 13

[Albert1]

A=$\underbrace{22--22}\,?\,\underbrace{33--33}$
$\,\,\,\,\,\, \,\,\,\,\, \,\,\, \,\, \,\, 50$$\,\,\,\,\,\,\,\, \,\,\,\,\, \,\,\,\,\,\, \,\, 50$
find $?$
(1) to make A a multiple of $7$
(2) to make A a multiple of $11$
(3) to make A a multiple of $13$

 

[Albert1]

A=$\underbrace{22--22}\,?\,\underbrace{33--33}$
$\,\,\,\,\,\, \,\,\,\,\, \,\,\, \,\, \,\, 50$$\,\,\,\,\,\,\,\, \,\,\,\,\, \,\,\,\,\,\, \,\, 50$
find $? (0\leq ? \leq 9)$
(1) to make A a multiple of $7$
(2) to make A a multiple of $11$
(3) to make A a multiple of $13$

hint :

Spoiler

$7|111111,\,\,11|111111, \, and \,\, 13|111111$

 

[kaliprasad]

A=$\underbrace{22--22}\,?\,\underbrace{33--33}$
$\,\,\,\,\,\, \,\,\,\,\, \,\,\, \,\, \,\, 50$$\,\,\,\,\,\,\,\, \,\,\,\,\, \,\,\,\,\,\, \,\, 50$
find $?$
(1) to make A a multiple of $7$
(2) to make A a multiple of $11$
(3) to make A a multiple of $13$

Spoiler

we have xxxxxx is divisible by 1001 or 7 * 11 * 13
so we remove 48 2's from A that is subtract 48 $2's * 10^{53}$ and then subtract {3 ...3}( 48 3's) and divide by 10^{48} and we get 22?33 . we have 22?22 divisible by 22 subtracting get
?11 . we can check that(by putting from 0 to 9) ? is 0 to be divisible by 11. 5 to be divisible by 7 and 6 to be divisible by 13

 

[Albert1]

Spoiler

we have xxxxxx is divisible by 1001 or 7 * 11 * 13
so we remove 48 2's from A that is subtract 48 $2's * 10^{53}$ and then subtract {3 ...3}( 48 3's) and divide by 10^{48} and we get 22?33 . we have 22?22 divisible by 22 subtracting get
?11 . we can check that(by putting from 0 to 9) ? is 0 to be divisible by 11. 5 to be divisible by 7 and 6 to be divisible by 13

nice !