Question

Number of natural numbers between 250 and 800 which are divisible by 7 is (a) 80 (b) 79 (c) 63 (d) 70

Short answer

Answer: (b) 79

Step-by-step solution

Find the first divisible natural number in the given range

Since we want to find the first natural number divisible by 7 within the given range (250 to 800), we can simply divide 250 by 7 and then round up to the nearest whole number. Afterward, we multiply that whole number by 7 to get the first multiple of 7 in the range. So, \(\lceil\frac{250}{7}\rceil\cdot7\), rounding up 250/7 results in 36, and 36*7 gets us 252. The first divisible natural number in the range is 252.

Find the last divisible natural number in the given range

To find the last divisible natural number within the given range (250 to 800), we can divide 800 by 7 and then round down to the nearest whole number. Afterward, we multiply that whole number by 7 to get the last multiple of 7 in the range. So, \(\lfloor\frac{800}{7}\rfloor\cdot7\), rounding down 800/7 results in 114, and 114*7 gets us 798. The last divisible natural number in the range is 798.

Find the total number of divisible natural numbers between first and last numbers

Now that we have found the first and the last divisible natural numbers in the given range (252 and 798), we can use these numbers to calculate the total number of natural numbers divisible by 7 in the given range. To find the number of terms divisible by 7 between 252 and 798, we can use the formula for the last term of an arithmetic series: \(a_n = a_1 + (n-1)d\), where \(a_n\) is the last term (798), \(a_1\) is the first term (252), n is the number of terms, and d is the common difference (7 in this case). Rearranging the formula to find n: \(n = \frac{a_n - a_1}{d} + 1\) \(n = \frac{798 - 252}{7} + 1\) \(n = \frac{546}{7} + 1\) \(n = 78 + 1\) \(n = 79\)

Choose the correct answer

There are 79 natural numbers between 250 and 800 that are divisible by 7. So, among the given options, answer choice (b) is the correct one.

Key concepts

Natural Numbers

Natural numbers, often denoted by the symbol \( \mathbb{N} \), are the set of positive integers that we use in everyday counting. They start from 1 and go on indefinitely - 1, 2, 3, 4, 5, and so on. Zero is often not considered a natural number when it comes to multiplication and division because dividing by zero is undefined, and multiplying any number with zero results in zero, which is not a part of the natural number set.

In our exercise, the task involved finding natural numbers between two larger numbers that are divisible by 7, illustrating how we can leverage arithmetic properties of natural numbers in practical applications. Understanding natural numbers is crucial because they form the foundation of more complex mathematical concepts such as arithmetic series and divisibility rules.

Arithmetic Series

An arithmetic series is a sequence of numbers in which each term after the first is obtained by adding a constant, known as the common difference \(d\), to the previous term. This type of series could represent, for instance, the total amount of an item collected over several days, if the same quantity is gathered each day. The series is often written as \(a_1, a_1 + d, a_1 + 2d, \) and so on, where \(a_1\) is the first term.

In the context of our problem, we used the properties of an arithmetic series to determine the total number of natural numbers within a range that are divisible by 7. The common difference in this case was 7, as numbers divisible by 7 are separated by multiples of 7. By using the formula for the nth term of an arithmetic series, students can easily calculate the number of terms within any specified range when the series is arithmetic in nature.

Divisibility Rules

Divisibility rules allow us to quickly determine if a number is divisible by another without performing long division. For example, a number is divisible by 2 if its last digit is even, it's divisible by 3 if the sum of its digits is divisible by 3, and so on for other divisors.

In our exercise, we applied divisibility by 7, which doesn't have a straightforward rule like smaller numbers but can still be checked using a process of elimination or using known multiples of 7 as a reference point. Understanding the divisibility rules is essential for students because they simplify calculations and can help efficiently solve problems involving divisibility criteria, especially when working with natural numbers and series.