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Chapter 1: Problem 5

Check whether \(6^{n}\) can end with the digit 0 for any natural number \(n\).

Short Answer

Expert verified
\(6^{n}\) cannot end with the digit 0 for any natural number \(n\) because its prime factorization does not include a factor of 5, which is necessary to be a multiple of 10.

Step by step solution

01

Understanding the Problem

A number ends with the digit 0 if it is a multiple of 10. To be a multiple of 10, a number must have factors of both 5 and 2.
02

Analyzing the Base

The number \(6^{n}\) has 6 as its base. The prime factorization of 6 is \(2 \times 3\). It does not contain the factor of 5.
03

Raising 6 to Any Power

Raising 6 to any power (\(n\)) will include the factors of 6's prime factorization, i.e., 2 and 3, raised to the power \(n\), which still does not introduce the factor of 5 needed to end with a 0.
04

Conclusion

Since the prime factorization of \(6^{n}\) for any natural number \(n\) will never include the factor 5, \(6^{n}\) cannot end with the digit 0 for any natural number \(n\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Numbers
Natural numbers are the most fundamental set of numbers used in mathematics. They consist of the positive integers starting from 1 and going onwards indefinitely (1, 2, 3, 4,鈥�).

In the context of the exercise, natural numbers play a role in determining the powers to which the base, 6, is raised. It's crucial to note that natural numbers do not include zero or negative numbers, which ensures that when we talk about raising 6 to the power of a natural number, we're starting from a baseline of at least 6 itself.

From a broader perspective, understanding natural numbers is essential for basic arithmetic operations, number theory, and many fields of mathematics where quantifying items in whole numbers is required.
Powers and Exponents
Powers and exponents denote how many times a number, known as the base, is multiplied by itself. An expression like \(6^n\) shows that 6 is the base and it鈥檚 raised to the power of \(n\), where \(n\) is the exponent.

Understanding Exponential Growth

As the value of \(n\) increases, the product also grows exponentially. This growth isn't linear but multiplies the base by itself \(n\) times.

Impact on Prime Factorization

In our exercise, the exponents only serve to increase the quantities of the prime factors within the base number. This concept is fundamental to understanding why raising 6 (which is 2 times 3) to any power will not result in a number ending with 0 without the presence of the prime factor 5.
Divisibility Rules
Divisibility rules help us determine if a number can be divided by another without leaving a remainder. For example, a number ends in 0 if, and only if, it is divisible by 10. To be divisible by 10, the number must include both 2 and 5 in its prime factorization.

Other Key Divisibility Rules

A number is divisible by:
  • 2 if the last digit is even (0, 2, 4, 6, 8).
  • 3 if the sum of its digits is divisible by 3.
  • 5 if the last digit is 0 or 5.

Applying these rules, we understand that no matter the power to which 6 is raised, it will never result in a number with a zero at the end, as the prime factor 5 is missing, which is a condition based on the divisibility by 10.

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Most popular questions from this chapter

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non- terminating repeating decimal expansion: (i) \(\frac{13}{3125}\) (ii) \(\frac{17}{8}\) (iii) \(\frac{64}{455}\) (iv) \(\frac{15}{1600}\) (v) \(\frac{29}{343}\) (vi) \(\frac{23}{2^{3} 5^{2}}\) (vii) \(\frac{129}{2^{2} 5^{7} 7^{5}}\) (viii) \(\frac{6}{15}\) (ix) \(\frac{35}{50}\) (x) \(\frac{77}{210}\)

An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups are to march in the same number of columns. What is the maximum number of columns in which they can march?

Use Euclid's division algorithm to find the HCF of : (i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255

Explain why \(7 \times 11 \times 13+13\) and \(7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1+5\) are composite numbers.

Show that any positive odd integer is of the form \(6 q+1\), or \(6 q+3\), or \(6 q+5\), where \(q\) is some integer.
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