show that 3n×4n cannot end with digit 0 or 5
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We have to prove that 3n x 4m cannot end with 0 and 5.
For the number to end with 0 it is necessary for the number to have 2 and 5 as its factors So,
2^2 x 5^2 = 100
2 x 5 = 10
Similarly for any number to end with 5 it needs to have 5 as one of its factors So,
5 x 6 = 30
3^n x 2^2m = 3^n x (2^2)m
As we can observe this number doesn't have 5 or 5 and 2 as its factors.
Therefore, the number does not end with 5 and 0.
Hence Proved.
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