sanjoy18 wrote:what is the reminder when 3^444+4^333 is divided by 5
A)0
B)1
C)2
D)3
E)4
When an integer with a units digit of 3 is raised to successive powers, the units digits of the resulting integers repeat in a cycle of 4:
3, 9, 7, 1...3, 9, 7, 1...
Implication: when the exponent is a multiple of 4, the units digit of the resulting integer will be 1.
This, 3^444 has a units digit of 1.
When an integer with a units digit of 4 is raised to successive powers, the units digits of the resulting integers alternate between 4 and 6:
4, 6...4, 6...
Implication: when the exponent is odd, the units digit of the resulting integer will be 4.
Thus, 4^333 has a units digit of 4.
Thus, 3^444 + 4^333 = integer with a units digit of 1 + integer with a units digit of 4 = integer with a units digit of 5.
An integer with a units digit of 5 is a multiple of 5.
Since 3^444 + 4^333 is a multiple of 5, dividing it by 5 will yield a remainder of 0.
The correct answer is
A.
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