check whether 6n can end with digit 0 for any natural number 'n'.

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Maths Hacker

Maths Hacker

Күн бұрын

To determine whether 6n can end with the digit 0 for any natural number n, let's analyze the problem:
1. Understanding the Problem:
o 6n represents the product of 6 and n, where n is any natural number (n∈Nn \in \mathbb{N}n∈N).
o A number ending with 0 must be divisible by 10. In other words, it must have both 2 and 5 as factors because 10=2×510 = 2 \times 510=2×5.
2. Factorization of 6:
o 6=2×36 = 2 \times 36=2×3. The number 6 includes 2 as a factor, but it does not include 5.
3. Product 6n6n6n:
o The product 6n will end with 0 only if it is divisible by 10. For 6n to be divisible by 10, n must contribute a factor of 5.
o However, since n is a natural number, n can be any positive integer. If n does not have 5 as a factor, 6n cannot end with 0.
4. Conclusion:
o No matter the value of n , 6n will always include 2 as a factor (from 6), but it will only include 5 as a factor if n specifically contains 5.
o However, n being a natural number and 6n being divisible by 10 would mean n has to be a multiple of 5. Even then, the last digit is determined by the cycles of multiplication involving 6, which does not inherently lead to 0 in this scenario.
Therefore, 6n cannot end with the digit 0 for any natural number n .
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