Question 26 of 51intermediate🔧 ApplyNumerical4 marks

How many 2-digit numbers are divisible by 3? What is the sum of all these 2-digit numbers?

Correct Answer

(Number of 2-digit numbers divisible by 3) 30

(Sum of all these 2-digit numbers) 1665

Exercise: Exercise Set 8.2 | Q: 5 | (Page 13)
For More Understanding

Explanation

Following the textbook's hint for similar problems, all 2-digit numbers divisible by 3 form an Arithmetic Progression (AP). The smallest and largest 2-digit numbers divisible by 3 are 12 and 99, respectively, giving us the first term, common difference, and last term of the AP. We find the number of terms using the nth term formula. To find the sum, we apply the method described in the context from Āryabhaṭiya: taking the sum of the first and last terms, divided by two (the average), and multiplied by the number of terms.

Solution Steps

  1. Step 1: Identify the AP. The 2-digit numbers divisible by 3 are 12, 15, 18, ..., 99. Here, first term (a) = 12, common difference (d) = 3, and last term (l) = 99.

  2. Step 2: Find the number of terms (n). Using the formula a_n = a + (n - 1)d, we substitute the known values: 99 = 12 + (n - 1)3.

  3. Step 3: Solve for n. This simplifies to 87 = 3(n - 1), which gives n - 1 = 29. Therefore, n = 30.

  4. Step 4: Calculate the sum. Using the sum formula from the context, Sum = (n/2) × (a + l). Substituting the values: Sum = (30/2) × (12 + 99) = 15×11115 \times 111 = 1665.

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