5*3523 is exactly divisible by 13 and 77. Fine the digit represent by *.

Answer :

We have a number 5x3523
If a number is divisible by 13 and 77  , than that number also divisible by  13  , 7 and 11

We know divisibility rule of 7 : Multiply each digit ( from right to left) by the digit in the corresponding position in this pattern (from left to right): 1, 3, 2, -1, -3, -2 (repeating for digits beyond the hundred-thousands place). Then sum the results.
So,  we get

5 ( - 2 ) + x ( - 3 ) + 3 ( - 1 )  + 5 ( 2 ) + 2 ( 3 ) + 3 ( 1 ) = 0

- 10 - 3x  - 3 + 10 + 6 + 3  = 0

3x  = 6

x  =  2  ,
So our number could be 523523 ,

Now we check that number 523523 is divisible by 11 or not

We know divisibility rule for 11 : If the number of digits is even, add the first and subtract the last digit from the rest.
And 523523 have even ( 6 ) digits , So  2352 + 5 - 3  =  2354  =  35 + 2 - 4   =  33 , and we know 33 is divisible by 11
So,
523523 is also divisible by 11

Now we check for 13
We know divisibility rule for 13 : Form the alternating sum of blocks of three from right to left.
So,  523523  =  523 - 523  = 0  , So 523523 is divisible by 13
So,

x  = 3                                                                                ( Ans )