# 60ab57377 is divisible by 99 .Find a and b?

In the given question we know that the number 60ab57377 is divisible by 99. We need to find the numbers a and b.

Here, use the fact that a number which is divisible by 99 is also divisible by 9 and 11 separately.

Now, the number 60ab57377 is divisible by 9.
Using the divisibility test, we know that if a number is divisible by 9, then the sum of all the digits of the number should be a multiple of 9. So,

Now, as a and b are single digit numbers, so they cannot take values greater than 9. So,

$a+b+35=36\phantom{\rule{0ex}{0ex}}a+b=36-35$
$a+b=1$             ... (1)

or

$a+b+35=45\phantom{\rule{0ex}{0ex}}a+b=45-35$
$a+b=10$         ... (2)
Similarly the number 60ab57377 is divisible by 11.
We know that if a number is divisible by 11, then the sum of the alternate digits minus the sum of the remaining digits is a multiple of 11. So,

Now, as a and b are single digit numbers, so they cannot take values greater than 9. So,

$a-b+7=11\phantom{\rule{0ex}{0ex}}a-b=11-7$
$a-b=4$  ... (3)

or

$a-b+7=0$
$a-b=-7$    ... (4)
Adding equation (2) and (3), we get

$a+b+a-b=10+4\phantom{\rule{0ex}{0ex}}2a=14\phantom{\rule{0ex}{0ex}}a=\frac{14}{2}\phantom{\rule{0ex}{0ex}}a=7$

Substituting $a=7$ in equation (2), we get

$a+b=10\phantom{\rule{0ex}{0ex}}7+b=10\phantom{\rule{0ex}{0ex}}b=10-7\phantom{\rule{0ex}{0ex}}b=3$

Therefore,

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