Show that any number of the form 75n,n€n can never end with the digit 0
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To show that any number of the form can never end with the digit :
Method of Induction.
Let the statement be
can never end with the digit .
Step 1. Let the statement be true for .
Then, does not end with .
Step 2. Let the statement be true for .
Then, does not end with the digit .
Step 3.
Now,
Since the statement is true for and , the statement is also true for .
Thus, any number of the form cannot end with .
Alternate Proof.
- We must know that any number that ends with the digit , also ends with the digit when takes a power.
- That is, ends with .
- The given number cannot end with the digit because any integral power of ends with .
- For example, , , and so on.
- We can see that they end with .
Thus, any number of the form cannot end with .
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