Question 39 of 51intermediate🔧 ApplyNumerical4 marks

How many three-digit numbers are divisible by 7?

Correct Answer

128

Exercise: END-OF-CHAPTER EXERCISES | Q: 3 | (Page 22)
For More Understanding

Explanation

The three-digit numbers divisible by 7 form an Arithmetic Progression (AP). The smallest three-digit number divisible by 7 is 105 (first term), and the largest is 994 (last term), with a common difference of 7. By applying the nth term formula of an AP, we calculate the total number of terms to be 128.

Solution Steps

  1. Step 1: Find the smallest three-digit number divisible by 7. The smallest 3-digit number is 100. Dividing 100 by 7 gives 14 with a remainder of 2. The next multiple of 7 is 15×715 \times 7 = 105. So, the first term a = 105.

  2. Step 2: Find the largest three-digit number divisible by 7. The largest 3-digit number is 999. Dividing 999 by 7 gives 142 with a remainder of 5. The largest multiple of 7 is 142×7142 \times 7 = 994. So, the last term a_n = 994.

  3. Step 3: Identify the common difference of the AP. The common difference d = 7.

  4. Step 4: Use the nth term formula to find the number of terms n. The formula is a_n = a + (n - 1)d. Substituting the values: 994 = 105 + (n - 1)7.

  5. Step 5: Solve for n. 994 - 105 = (n - 1)7 => 889 = (n - 1)7 => n - 1 = 127 => n = 128.

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