vaswani.sharan wrote:What is the remainder when 43^43+ 33^33 is divided by 10?
When a positive integer is divided by ten, the remainder is the UNITS DIGIT of the integer.
To illustrate:
25/10 = 2 R5.
Thus, we need to determine the units digit of 43�³ + 33³³.
We can treat this as a PATTERN question.
The strategy: WRITE IT OUT until you see the pattern.
3¹ = 3.
3² = 9.
3³ = 27.
3� = 81.
3� = 243.
Etc.
The units digits repeat in a CYCLE OF 4: 3,9,7,1...3,9,7,1...
Thus, when a positive integer with a units digit of 3 is raised to a power that is a multiple of 4 -- as in 43� -- the units digit will be 1.
From there, the pattern will repeat: 3,9,7,1...3,9,7,1...
Units digit of 43�³:
43�� has a unit digit of 1, since the exponent is a multiple of 4.
From here, the cycle repeats:
43�¹ has a units digit of 3.
43�² has a units digit of 9.
43�³ has a units digit of 7.
Units digit of 33³³:
33³² has a unit digit of 1, since the exponent is a multiple of 4.
From here, the cycle repeats:
33³³ has a units digit of 3.
Since 43�³ has a units digit of 7, 33³³ has a units digit of 3, and 7+3 = 10, the units digit of 43�³ + 33³³ is 0.
Thus, the remainder when 43�³ + 33³³ is divided by 10 will be 0.
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