Question

The value of \((37)^{3^{x}}-(33)^{3^{x}}\) ends in \((x \in N)\). (1) 4 (2) 6 (3) 0 (4) Either (1) or (2)

Short answer

Based on the given information and the analysis, the correct answer is: (4) Either (1) or (2) This is because the last digit of the expression \((37)^{3^{x}}-(33)^{3^{x}}\) alternates between 4 and 6 for x belonging to natural numbers.

Step-by-step solution

Analyze the pattern of the last digit of \((37)^{3^{x}}\) and \((33)^{3^{x}}\).

To analyze the pattern, let's write down the last digit of the first few terms in the sequence \((37)^{3^{x}}\) and \((33)^{3^{x}}\). For \((37)^{3^x}\), we have: x = 0 : last digit of \((37)^{3^0} = (37)^{1} = 7\) x = 1 : last digit of \((37)^{3^1} = (37)^{3} = 3\) x = 2 : last digit of \((37)^{3^2} = (37)^{9} = 7\) and so on. Notice that the last digit repeats in a cycle after every 2 terms, i.e., 7, 3, 7, 3, ... For \((33)^{3^x}\), we have: x = 0 : last digit of \((33)^{3^0} = (33)^{1} = 3\) x = 1 : last digit of \((33)^{3^1} = (33)^{3} = 7\) x = 2 : last digit of \((33)^{3^2} = (33)^{9} = 3\) and so on. For \((33)^{3^x}\), the last digit also repeats in a cycle after every 2 terms, i.e., 3, 7, 3, 7, ...

Subtract the last digits of the sequences \((37)^{3^{x}}\) and \((33)^{3^{x}}\).

Now, we will subtract the last digits of the two sequences to find the last digit of the given expression: 7 - 3 = 4 3 - 7 = -4 (which is equivalent to 6, since we are only interested in the last digit)

Identify the correct option

Since the last digit of the expression \((37)^{3^{x}}-(33)^{3^{x}}\) alternates between 4 and 6, the correct answer is: (4) Either (1) or (2)

Key concepts

Number Theory

In mathematics, number theory is a branch of pure mathematics concerned with the properties of numbers, particularly integers. It delves into numbers and the relationships between them, often using patterns and algebraic structures to uncover new truths. In the context of our exercise, number theory helps us understand the behavior of the last digits of powers of integers. This understanding is essential for tackling problems like the one involving \(37^{3^{x}}\) and \(33^{3^{x}}\).

For instance, when studying the exponentiation of a number ending in 7 or 3, we realize that there's a pattern to the last digit that emerges, irrespective of the size of the exponent. These patterns are predictable and cyclic, which is incredibly useful for finding the last digit of large exponentiations without having to calculate the entire power. Number theory, therefore, provides us with tools and concepts such as divisibility, modular arithmetic, and the notion of congruences, which can simplify complex problems into manageable calculations.

Cycles in Exponents

Cycles in exponents refer to the recurring pattern in the last digits of numbers raised to various powers. As demonstrated in the solution, the numbers 7 and 3 have an interesting property where their powers end in a limited set of digits that repeat in a cycle. For instance, any number ending in 7, when raised to a power, will have its last digit cycle through a specific sequence, repeating after every few exponents.

Recognizing the cyclical nature of exponents allows us to 'mod out' the cycle's length and directly determine the last digit for very large powers without extensive computation. In our example, the cycle for \(37^{3^{x}}\) is 7, 3 and for \(33^{3^{x}}\), it's 3, 7. Understanding these cycles is pivotal in number theory and can significantly simplify seemingly daunting tasks involving exponentiation. This is an excellent example of the practical application of cycles in exponents to solve real mathematical problems.

Problem Solving Techniques

Identifying and Exploiting Patterns

One critical problem-solving technique in mathematics is to identify and exploit patterns. As shown in the problem's solution, recognizing the cyclic pattern in the last digits of exponentiation can lead to quick and effective solutions. By doing this, we avoid lengthy and sometimes impossible calculations.

Work with Simplified Cases

Another technique is to work with simplified cases. For example, examining the behavior of small powers can reveal patterns that apply to larger, more complex cases. This is what we did when observing the last digit of \(37^{3^x}\) for small values of x. This problem-solving approach can sometimes generalize to broader situations.

Modular Arithmetic

Finally, the use of modular arithmetic is a powerful tool in problem solving. By focusing only on the last digits (which involves working mod 10), we reduce the problem to one of simple arithmetic rather than laborious exponentiation. Modular arithmetic allows us to work within a 'residue system' where computations are significantly more accessible and reveal much about the properties of the numbers involved.