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.

Step 1: Understand the condition for a number to end in zero

For any integer to end with the digit zero, it must be divisible by 10. Since $10 = 2 \times 5$, this means that its prime factorization must contain at least one factor of 2 and at least one factor of 5.

Step 2: Find the prime factorization of the base number

First, we find the prime factorization of the base number, which is 4. The prime factors of 4 are $2 \times 2 = 2^2$. This is a fundamental step as per the Fundamental Theorem of Arithmetic.

Step 3: Express $4^n$ in terms of its prime factors

Now we substitute the prime factorization of 4 into the expression $4^n$. Using the exponent rule $(a^m)^n = a^{mn}$, we find that $4^n$ can be written as $2^{2n}$. This shows the prime factors of $4^n$.

Step 4: Analyze the prime factors of $4^n$

From the expression $2^{2n}$, it is clear that the only prime factor of $4^n$ is 2. The Fundamental Theorem of Arithmetic states that the prime factorization of any natural number is unique. Therefore, there are no other prime factors present. As per Reference 3, this understanding is critical.

Step 5: Conclusion based on prime factorization

Since the prime factor 5 is not present in the prime factorization of $4^n$, it cannot be divisible by 5. Consequently, $4^n$ cannot end with the digit zero for any natural number $n$, as a number ending in zero must have both 2 and 5 as prime factors. This aligns with the example given in Reference 1 of NCERT Class 10 Mathematics.

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