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Determine the unit digit in the expansion of the following: (4987)184
- a)3
- b)7
- c)9
- d)1
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
Determine the unit digit in the expansion of the following: (4987)184a...
The given number is (4987)184
⇒ the unit digit of the given number will be the unit digit of number raised to 184 i.e., 7184
⇒ 71 = 7, 72 = 49 (unit digit is 9); 73 = 343 (unit digit is 3); 74 = 2401(unit digit 1)
⇒ 75 = 16807 (unit digit 7 again)
∴ with 7 as the unit digit raised to power, the only possibilities of unit digits are 7, 9, 3 and 1.
⇒ 7184 can be written as 7(4 × 46)
⇒ (74)46 ⇒ (1)46 = 1
∴ (4987)184 will have 1 as its unit digit.
⇒ the unit digit of the given number will be the unit digit of number raised to 184 i.e., 7184
⇒ 71 = 7, 72 = 49 (unit digit is 9); 73 = 343 (unit digit is 3); 74 = 2401(unit digit 1)
⇒ 75 = 16807 (unit digit 7 again)
∴ with 7 as the unit digit raised to power, the only possibilities of unit digits are 7, 9, 3 and 1.
⇒ 7184 can be written as 7(4 × 46)
⇒ (74)46 ⇒ (1)46 = 1
∴ (4987)184 will have 1 as its unit digit.
Most Upvoted Answer
Determine the unit digit in the expansion of the following: (4987)184a...
Understanding the Problem
To find the unit digit of (4987)^184, we need to focus on the unit digit of the base number, which is 7 in this case.
Identifying the Unit Digit Pattern
The unit digits of powers of 7 form a repeating cycle:
- 7^1 = 7 (Unit digit is 7)
- 7^2 = 49 (Unit digit is 9)
- 7^3 = 343 (Unit digit is 3)
- 7^4 = 2401 (Unit digit is 1)
After 4 terms, the pattern repeats:
- 7^5 = 7 (Unit digit is 7)
- 7^6 = 49 (Unit digit is 9)
Finding the Relevant Power
Now, we need to determine which unit digit corresponds to 7^184. To do this, we calculate 184 modulo 4 (since the cycle length is 4):
- 184 ÷ 4 = 46 with a remainder of 0.
Since the remainder is 0, it indicates we are at the end of the cycle, which corresponds to 7^4.
Conclusion: The Unit Digit
From our earlier pattern, the unit digit of 7^4 is 1. Therefore, the unit digit of (4987)^184 is:
- 1
Thus, the correct answer is option D: 1.
To find the unit digit of (4987)^184, we need to focus on the unit digit of the base number, which is 7 in this case.
Identifying the Unit Digit Pattern
The unit digits of powers of 7 form a repeating cycle:
- 7^1 = 7 (Unit digit is 7)
- 7^2 = 49 (Unit digit is 9)
- 7^3 = 343 (Unit digit is 3)
- 7^4 = 2401 (Unit digit is 1)
After 4 terms, the pattern repeats:
- 7^5 = 7 (Unit digit is 7)
- 7^6 = 49 (Unit digit is 9)
Finding the Relevant Power
Now, we need to determine which unit digit corresponds to 7^184. To do this, we calculate 184 modulo 4 (since the cycle length is 4):
- 184 ÷ 4 = 46 with a remainder of 0.
Since the remainder is 0, it indicates we are at the end of the cycle, which corresponds to 7^4.
Conclusion: The Unit Digit
From our earlier pattern, the unit digit of 7^4 is 1. Therefore, the unit digit of (4987)^184 is:
- 1
Thus, the correct answer is option D: 1.
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Determine the unit digit in the expansion of the following: (4987)184a)3b)7c)9d)1Correct answer is option 'D'. Can you explain this answer?
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