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vaibhav101
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When an integer is raised to consecutive powers, the resulting units digits repeat in a CYCLE.vaibhav101 wrote:find the units digit in $$57867^{192567}-1452^{876}$$ .
A 3
B 6
C 5
D 7
E 1
57867¹�²���:
7¹ --> units digit of 7.
7² --> units digit of 9. (Since the product of the preceding units digit and 7 = 7*7 = 49.)
7³ --> units digit of 3. (Since the product of the preceding units digit and 7 = 9*7 = 63.)
7� --> units digit of 1. (Since the product of the preceding units digit and 7 = 3*7 = 21.)
From here, the units digits will repeat in the same pattern: 7, 9, 3, 1.
The units digits repeat in a CYCLE OF 4.
Implication:
When an integer with a units digit of 7 is raised to a power that is a multiple of 4, the units digit will be 1.
From there, the cycle of units digits will repeat: 7, 9, 3, 1.
192576/4 = 48141 R3.
The result in blue implies that 57867¹�²��� will go through 48141 cycles of 4 (yielding a units digit of 1) and then 3 more places in the unit-digit cycle:
7, 9, 3.
Thus, 57867¹�²��� has a units digit of 3.
1452���:
2¹ --> units digit of 2.
2² --> units digit of 4.
2³ --> units digit of 8.
2� --> units digit of 6. (Since the product of the preceding units digit and 2 = 8*2 = 16.)
From here, the units digits will repeat in the same pattern: 2, 4, 8, 6.
The units digit repeat in a CYCLE OF 4.
Implication:
When an integer with a units digit of 2 is raised to a power that is a multiple of 4, the units digit will be 6.
876 is a multiple of 4 because its last two digits -- 76 -- form an integer that can divided twice by 2.
Since 1452��� is raised to a power that is a multiple of 4, it will yield a units digit of 6.
When an integer with units digit of 6 is subtracted from an integer with a units digit of 3, the result has a units digit of 7:
53 - 36 = 17.
Thus:
57867¹�²��� - 1452��� = (integer with a units digit of 3) - (integer with a units digit of 6) = integer with a units digit of 7.
The correct answer is D.
