# GMAT Math Question | Progressions Q17

#### GMAT Question Bank | Arithmetic Sequence & Progressions

The GMAT Math practice question given below is a sequences and series question based on Arithmetic Progressions about finding number of terms of an Arithmetic sequence. An easy question - GMAT 600 to 650 level problem solving question.

Question 17: How many 3 digit positive integers exist that when divided by 7 leave a remainder of 5?

1. 128
2. 142
3. 143
4. 141
5. 129

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### Explanatory Answer

#### Step 1 of solving this GMAT Arithmetic Progressions question: Find the first and last term of the series

The smallest 3-digit positive integer that leaves a remainder of 5 when divided by 7 is 103.
The largest 3-digit positive integer that leaves a remainder of 5 when divided by 7 is 999.

The series of numbers that satisfy the condition that the number should leave a remainder of 5 when divided by 7 is an A.P (arithmetic progression) with 103 as the first term and 999 as the last term. The common difference of the sequence is 7.

#### Step 2 of solving this GMAT AP question: Compute the number of terms

In an A.P, the last term l = a + (n - 1) * d, where 'a' is the first term, 'n' is the number of terms of the series and 'd' is the common difference.

Therefore, 999 = 103 + (n - 1) * 7
Or 999 - 103 = (n - 1) * 7
Or 896 = (n - 1) * 7
So, n - 1 = 128 or n = 129

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