Question

For each \(n \in N, 2^{3 n}-7 n-1\) is divisible by (a) 64 (b) 36 (c) 49 (d) 25

Short answer

After a step-by-step analysis, it can be concluded that the given expression \(2^{3n} - 7n - 1\) is not consistently divisible by any of the given options (a) 64, (b) 36, (c) 49, or (d) 25 for all natural numbers \(n\).

Step-by-step solution

Observe powers of 2

Notice that for every natural number \(n\), the term \(2^{3n}\) will always be a power of 2 (an even number). Therefore, the difference \(2^{3n} - 7n\) will also be an even number. Subtracting 1 from an even number gives an odd number. Hence, \(2^{3n} - 7n - 1\) will always be odd.

Eliminate options that are even

Since \(2^{3n} - 7n - 1\) is always odd, it cannot be divisible by any even number. In the given options, 64 and 36 are even numbers. Therefore, they can be eliminated. We are now left with two options: (c) 49 and (d) 25.

Test for smaller values of n

We can test the divisibility for smaller values of \(n\) for the remaining options. However, we should first try \(n=1\) and see which options satisfy the condition. \(2^{3(1)} - 7(1) - 1 = 8 - 7 - 1 = 0\) This shows that the expression is not divisible by either 49 or 25 when \(n = 1\). Now, let's try \(n = 2\). \(2^{3(2)} - 7(2) - 1 = 64 - 14 - 1 = 49\) This shows that the expression is divisible by 49 but not 25 for \(n = 2\). Therefore, we can conclude that the expression is divisible by 49 for some values of \(n\). However, we should check more cases before deciding which option is correct.

Analyze the divisibility by 49 for larger values of n

We can rewrite the given expression as follows: \(2^{3n} - 7n - 1 = 2^{3n} - (32n + 25n) - 1 = 2^{3n} - 32n - 1 - 25n\) Now, observe that when \(n\) is odd, \(2^{3n}\) is a power of 2, and hence, an even number. Furthermore, the term \(32n\) will also be even, as 32 is even. However, the term \(25n\) will be odd since 25 is odd. Therefore, when we add the terms, their sum will be odd. Subtracting 1 from an odd number gives an even number. Since 49 is odd, the expression cannot be divisible by 49 when \(n\) is odd. Therefore, we can conclude that there is no consistent divisibility by either 25 or 49.

Conclusion

After step-by-step analysis, we can conclude that there is no consistent divisibility for the given options. Therefore, the given expression is not divisible by any of the given options (a) 64, (b) 36, (c) 49, or (d) 25 for all natural numbers \(n\).

Key concepts

Divisibility Rules

Divisibility rules help us quickly determine if a number is divisible by another without performing actual division. They are simple checks based on the properties of numbers. For instance:

• If a number ends in 0, then it is divisible by 10.
• If the sum of a number's digits is a multiple of 9, it is divisible by 9.
• Any even number (number ending in 0, 2, 4, 6, or 8) is divisible by 2.

In the context of this exercise, the expression is analyzed for divisibility by 64, 36, 49, and 25. The exercise shows that the expression is never consistently divisible by any of these numbers for all natural numbers, illustrating the importance of rules in narrowing down options for divisibility.

Even and Odd Numbers

Even and odd numbers are fundamental concepts in mathematics. An even number is any integer that can be divided by 2 without leaving a remainder. Examples include 0, 2, 4, 6, and 8. Odd numbers, conversely, leave a remainder of 1 when divided by 2. Examples are 1, 3, 5, 7, and 9.

In our exercise, understanding evenness and oddness is crucial. The term \(2^{3n}\) is even, and thus \(2^{3n} - 7n\) is also even. However, subtracting 1 turns it odd, i.e., \(2^{3n} - 7n - 1\) is always odd. This simple property led us to eliminate even divisors like 64 and 36 immediately.

Natural Numbers

Natural numbers are the set of positive integers starting from 1, and often include 0 as well, depending on the context. They are denoted by \(\mathbb{N}\). These numbers are used for counting and ordering.

In the problem presented, \( n \) is taken from the set of natural numbers. This assumption is critical because it limits \( n \) to positive whole numbers, excluding 0 when necessary. This property helps us test different values of \( n \) easily. For instance, using \( n = 1 \) or \( n = 2 \), which are straightforward, helps us quickly understand patterns or find divisibility by checking specific cases.

Exponents

Exponents represent repeated multiplication of a number, known as the base. For example, \( 2^3 \) (read as "2 raised to the power of 3") is calculated as \( 2 \times 2 \times 2 = 8 \).

In the expression \( 2^{3n} \), \( 2 \) is the base raised to the power of \( 3n \). This means the base is multiplied by itself \( 3n \) times. Understanding how exponents work is essential. Exponents grow numbers rapidly, impacting the overall behavior of expressions. In solutions like these, deciphering the behavior of terms like \( 2^{3n} \) as \( n \) changes is crucial to identify divisibility patterns, though the final verdict showed inconsistency in consistent divisibility by any particular given number for all \( n \).