Exercise 1.1, 5

Check whether 6ⁿ can end with the digit 0 for any natural number n.

Solution:

Condition for a number to end in 0

A number ends in 0 if and only if it is divisible by 10, which means it must be divisible by both 2 and 5.

Prime Factors of 6ⁿ

6ⁿ = (2 × 3)ⁿ = 2ⁿ × 3ⁿ

• The factorization shows that 6ⁿ consists of only powers of 2 and 3.
• It does not contain 5 as a factor.

Checking for divisibility by 10

• Since 6ⁿ is not divisible by 5, it can never be divisible by 10.
• Therefore, 6ⁿ can never end with the digit 0 for any natural number n.

Final Answer:

            No, 6ⁿ cannot end with the digit 0 for any natural number n.