Question

All numbers from 1 to 200 (in decimal system) are written in base 6 and base 7 systems. How many of the numbers will have a non-zero units digit in both base 6 and base 7 notations?

Explanatory Answer

If a number written in base 6 ends with a zero, it should be a multiple of 6. In other words, the question wants us to find all numbers from 1 to 200 that are not multiples of 6 or 7. There are 33 multiples of 6 less than 201. There are 28 multiples of 7 less than 201. There are 4 multiples of 6 & 7 (or multiple of 42) from 1 to 200.

So, total multiples of 6 or 7 less than 201 = 33 + 28 - 4 = 57. Number of numbers with non-zero units digit = 200 - 57 = 143.

Correct Answer : 143