Consider the numbers 4n, where n is a natural number. Check whether there is any value of n for which 4n ends with the digit zero.
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Step-by-Step Solution
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=2×5, 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 2×2=22. This is a fundamental step as per the Fundamental Theorem of Arithmetic.
Step 3: Express 4n in terms of its prime factors
Now we substitute the prime factorization of 4 into the expression 4n. Using the exponent rule (am)n=amn, we find that 4n can be written as 22n. This shows the prime factors of 4n.
Step 4: Analyze the prime factors of 4n
From the expression 22n, it is clear that the only prime factor of 4n 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 4n, it cannot be divisible by 5. Consequently, 4n 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.