show that 12n cannot end with digit 0 or 5 for any natural number n
Here, Number=12n where n stand for any natural number .
Now 12n= (22 x3)n
Now , For 12n to end with 0, it should have 2 as well as 5 in its Prime factors to end with 0, Also to end with 5 , it requires at least a single multiple of 5 in its Prime Factors, So 12n cannot end with the digit 0 or 5.
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By the law that a number can be expressed as the powers of its factors.
Therefore, any number ending with 0 or 5 must have its factors as 2n * 5n as 0 as unit digit is only possible by the interactions of powers of 2&5.
Since factors of 12 are 2 and 3. There is no 5 as factor.
Therefore 12n can't end with 0 or 5.
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