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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

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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.

 


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