NCERT Class X Chapter 1: Real Numbers Example 1
NCERT Class X Chapter 1: Real Numbers Example 1
Question:
Consider the numbers \(4^n\), where \(n\) is a natural number.
Check whether there is any value of \(n\) for which \(4^n\) ends with the digit zero.
Given:
The numbers considered are of the form \(4^n\), where \(n\) is a natural number (\(n \in \mathbb{N}\)).
To Find:
Determine whether there exists any natural number \(n\) such that \(4^n\) ends with the digit zero.
Formula:
To end with the digit zero, a number must be divisible by 10.
\(10 = 2 \times 5\)
Solution:
Step 1: State the condition for a number to end in zero.
A number ends with the digit zero if and only if it is divisible by 10. That is, it must have both 2 and 5 as its prime factors.
Step 2: Express \(4^n\) in terms of its prime factors.
We know that \(4 = 2^2\), so:
Step 3: Apply the exponent rule \((a^m)^n = a^{mn}\).
$$ (2^2)^n = 2^{2n} $$Step 4: Analyze the factors of \(4^n\).
\(2^{2n}\) contains only the prime factor 2, repeated \(2n\) times. There is no factor of 5 in \(4^n\) for any natural number \(n\).
Step 5: Conclude about divisibility by 10.
Since \(4^n\) does not have 5 as a factor for any \(n\), it cannot be divisible by 10. Therefore, \(4^n\) can never end with the digit zero.
Result:
No, there is no value of \(n\) for which \(4^n\) ends with the digit zero.
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