Mathematics — Standard (041) — CBSE Class 10 board question

No, $(15)^n$ cannot end with the digit 0.

For a number to end with 0, it must be divisible by 10, i.e., its prime factorisation must contain both 2 and 5.

Now, $15^n = (3 \times 5)^n = 3^n \times 5^n$.

The prime factorisation of $15^n$ contains only the primes 3 and 5 — it does not contain 2 as a factor. By the uniqueness of the Fundamental Theorem of Arithmetic, 2 can never appear in this factorisation. Hence, $15^n$ can never end with the digit 0.

Source: Chapter 1, Section 1.2 – The Fundamental Theorem of Arithmetic

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• The key idea: a number ends in 0 only if both 2 and 5 are factors (since $10 = 2 \times 5$).
• Show the prime factorisation of $15^n$ and note that 2 is absent.
• Invoke the uniqueness part of the Fundamental Theorem of Arithmetic — this is the formal justification examiners expect, not just an intuitive argument.
• This mirrors the textbook's Example 1 on $4^n$; follow the same reasoning pattern.