Check whether there is any natural number 'n' for which (14)n ends with the digit '0' or '5'.
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Step-by-Step Solution
Step 1: Understand the condition for ending in 0 or 5
For a natural number to end with the digit '0', its prime factorization must include both 2 and 5. For a natural number to end with the digit '5', its prime factorization must include 5.
Step 2: Find the prime factors of the base number
First, we find the prime factorization of the base number, which is 14. The prime factors of 14 are 2 and 7.
Step 3: Determine the prime factors of (14)n
Now, we determine the prime factors of (14)n. Using the property of exponents, (14)n can be written as (2×7)n, which simplifies to 2n×7n. This means the only prime factors of (14)n are 2 and 7, regardless of the value of n.
Step 4: Check for the presence of prime factor 5
Since the prime factorization of (14)n only contains the prime factors 2 and 7, it does not contain the prime factor 5. Therefore, (14)n cannot end with the digit '0' or '5' for any natural number n.