Question

An integer N, $100<N<200$, when expressed in base 5 notation has a final (i.e. rightmost) digit of 0. When N is expressed in base 8 notation and base 11 notation, the leading (i.e. leftmost) digit is 1 in both cases. Then N when divided by 7 gives a remainder of:
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Answer 1

Expressing in base 5 has final digit 0. So, it has to be a multiple of 5.
Expressing in base 8 notation has 1 as leftmost digit. So, it has to be (1xy) in base 8.
So, the number varies from 100 to 127 (177 in base 8 = 127 in base 10).
Expressing in base 11 notation, 1 is the left most digit. Again, it has to be (1ab) in base 11.
So, the integer is greater than 121 (100 in base 11 = 121 in base 10).
Combining three conditions, we get the number as 125.
$\therefore \left(\frac{125}{7}\right)$, remainder = 6.