How many numbers less than 904 are divisible by 3
The Correct Answer and Explanation is:
To determine how many numbers less than 904 are divisible by 3, we need to consider numbers in a sequence that are multiples of 3. Let’s break it down step by step:
- Identify the first and last multiples of 3 under 904:
The smallest positive integer divisible by 3 is 3. The largest integer divisible by 3 that is less than 904 can be found by dividing 904 by 3 and rounding down to the nearest whole number. [
904 \div 3 = 301.33
]
Since we’re looking for whole numbers, the largest multiple of 3 below 904 is 3 × 301 = 903. - Form a sequence of multiples of 3:
The sequence of numbers divisible by 3 below 904 is:
[
3, 6, 9, 12, \dots, 903
]
This is an arithmetic sequence where the first term (a = 3), the common difference (d = 3), and the last term (l = 903). - Use the formula for the number of terms in an arithmetic sequence:
The formula to calculate the number of terms (n) in an arithmetic sequence is:
[
n = \frac{l – a}{d} + 1
]
Substituting the known values:
[
n = \frac{903 – 3}{3} + 1
]
[
n = \frac{900}{3} + 1 = 300 + 1 = 301
]
Thus, the total number of numbers less than 904 that are divisible by 3 is 301.
Explanation:
Multiples of 3 are integers that result from multiplying 3 by other integers. By creating a sequence starting from the smallest multiple of 3 (which is 3 itself) and continuing to the largest multiple of 3 under 904 (which is 903), we form a sequence where each term increases by 3. The formula for the number of terms in this arithmetic sequence gives us a way to count how many such numbers exist between 3 and 903 inclusively. Since each step in the sequence adds 3, the numbers grow in a regular, predictable manner. Therefore, there are 301 numbers less than 904 that are divisible by 3.