CAT Quantitative Aptitude Questions | CAT Permutation and Combination Questions

The question is from Permutation and Combination. Another question which combines number theory with combinatorics. We need to find out how many numbers will have non-zero units digit when it is converted to another base system. This section hosts a number of questions which are on par with CAT questions in difficulty on CAT Permutation and Combination, and CAT Probability.

Question 9: 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?

1. 143
2. 200
3. 157
4. 122

Explanatory Answer

Method of solving this CAT Question from Permutation and Combination: A number in base n that ends in a zero is a multiple of n.

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.

The question is "How many of the numbers will have a non-zero units digit in both base 6 and base 7 notations?"

Hence the answer is "143"

Choice A is the correct answer.