Math, asked by prakashtarun320, 9 months ago

Show that any number of the form 75n,n€n can never end with the digit 0​

Answers

Answered by Swarup1998
4

To show that any number of the form \boxed{75^{n},n\in\mathbb{N}} can never end with the digit \boxed{0} :

Method of Induction.

Let the statement be

P(n):75^{n},n\mathbb{N} can never end with the digit 0.

Step 1. Let the statement be true for n=1.

Then, 75^{1}=75 does not end with 0.

Step 2. Let the statement be true for n=k.

Then, 75^{k} does not end with the digit 0.

Step 3.

Now, P(k+1)

=75^{k+1}

=75^{k}\times 75^{1}

=P(k)\times P(1)

Since the statement is true for n=1 and n=k, the statement is also true for n=k+1.

Thus, any number of the form 75^{n},n\in\mathbb{N} cannot end with 0.

Alternate Proof.

  • We must know that any number that ends with the digit 0, also ends with the digit 0 when takes a power.

  • That is, (A...0)^{n} ends with 0.

  • The given number 75^{n} cannot end with the digit 0 because any integral power (>0) of 5 ends with 5.

  • For example, 75^{1}=75, 75^{2}=5625, 75^{3}=421875 and so on.

  • We can see that they end with 5.

Thus, any number of the form 75^{n},n\in\mathbb{N} cannot end with 0.

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