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

We have a number 5

*x*3523

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 - 3

*x*- 3 + 10 + 6 + 3 = 0

3

*x*= 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 )
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