Number Properties - Counting Methods

Concept: Prime factorization & divisibility

The concept tested in this GMAT practice question is counting method and elementary number properties.

Question: How many different positive integers exist between 106 and 107, the sum of whose digits is equal to 2?

  1. 6
  2. 7
  3. 5
  4. 8
  5. 18

Explanatory Answer

Video explanation provided for this question
Find the number of such integers existing for a lower power of 10 and extrapolate the results.

Between 10 and 100, that is 101 and 102, we have 2 numbers, 11 and 20.
Similarly, between 100 and 1000, that is 102 and 103, we have 3 numbers, 101, 110 and 200.

Therefore, between 106 and 107, one will have 7 integers whose sum will be equal to 2.

Alternative approach

All numbers between 106 and 107 will be 7 digit numbers.
There are two possibilities if the sum of the digits has to be '2'.

Possibility 1: Two of the 7 digits are 1s and the remaining 5 are 0s.
The left most digit has to be one of the 1s. That leaves us with 6 places where the second 1 can appear.
So, a total of six 7-digit numbers comprising two 1s exist, sum of whose digits is '2'.

Possibility 2: One digit is 2 and the remaining are 0s.
The only possibility is 2000000.

Total count is the sum of the counts from these two possibilities = 6 + 1 = 7

Choice B is the correct answer.

Video Explanation

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