Find the sum of all two-digit natural numbers which are not divisible by 3.
The correct answer and explanation is:
To find the sum of all two-digit natural numbers which are not divisible by 3, follow these steps:
Step 1: Identify the Range
The two-digit natural numbers range from 10 to 99.
Step 2: Find the Sum of All Two-Digit Numbers
To find the sum of all two-digit numbers, use the formula for the sum of an arithmetic series: S=n2×(a+l)S = \frac{n}{2} \times (a + l)
Where:
- nn is the number of terms
- aa is the first term
- ll is the last term
For the numbers from 10 to 99:
- a=10a = 10
- l=99l = 99
- n=99−10+1=90n = 99 – 10 + 1 = 90
So the sum of all two-digit numbers is: S=902×(10+99)=45×109=4905S = \frac{90}{2} \times (10 + 99) = 45 \times 109 = 4905
Step 3: Find the Sum of Two-Digit Numbers Divisible by 3
To find the numbers divisible by 3, first identify the smallest and largest two-digit numbers divisible by 3:
- The smallest is 1212 (since 10÷3=3.3310 \div 3 = 3.33 and the next whole number is 12).
- The largest is 9999 (since 99÷3=3399 \div 3 = 33).
These numbers form an arithmetic sequence with:
- First term a=12a = 12
- Last term l=99l = 99
- Common difference d=3d = 3
The number of terms in this sequence is: n=99−123+1=30n = \frac{99 – 12}{3} + 1 = 30
So the sum of the numbers divisible by 3 is: S3=302×(12+99)=15×111=1665S_3 = \frac{30}{2} \times (12 + 99) = 15 \times 111 = 1665
Step 4: Find the Sum of Two-Digit Numbers Not Divisible by 3
Now, subtract the sum of numbers divisible by 3 from the total sum: Snot divisible by 3=4905−1665=3240S_{\text{not divisible by 3}} = 4905 – 1665 = 3240
Thus, the sum of all two-digit natural numbers which are not divisible by 3 is 3240.