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Find the unit digit of (432)412 × (499)431
- a)1
- b)3
- c)2
- d)5
- e)4
Correct answer is option 'E'. Can you explain this answer?
Most Upvoted Answer
Find the unit digit of (432)412 × (499)431a)1b)3c)2d)5e)4Correct...
To find the unit digit of (432)412, we need to find the remainder when (432)412 is divided by 10.
Let's first look at the unit digits of powers of 2:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16 (unit digit is 6)
2^5 = 32 (unit digit is 2)
2^6 = 64 (unit digit is 4)
2^7 = 128 (unit digit is 8)
2^8 = 256 (unit digit is 6)
We notice that the unit digit of 2^n repeats in cycles of 4: 2, 4, 8, 6.
Now, let's look at the exponent 412. We can write it as:
412 = 4(100) + 1(10) + 2
So we need to find the remainder when (432)412 is divided by 10, which is the same as finding the remainder when (432)4(100) * (432)1(10) * (432)2 is divided by 10.
Using the cycles of 2^n unit digits, we know that the unit digit of (432)4(100) is 6, and the unit digit of (432)1(10) is 2.
Now we just need to find the unit digit of (432)2.
(432)2 = 186624
The unit digit is 4.
So the remainder when (432)412 is divided by 10 is the same as the remainder when 6 * 2 * 4 is divided by 10.
6 * 2 * 4 = 48
The unit digit of 48 is 8.
Therefore, the unit digit of (432)412 is 8.
Let's first look at the unit digits of powers of 2:
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16 (unit digit is 6)
2^5 = 32 (unit digit is 2)
2^6 = 64 (unit digit is 4)
2^7 = 128 (unit digit is 8)
2^8 = 256 (unit digit is 6)
We notice that the unit digit of 2^n repeats in cycles of 4: 2, 4, 8, 6.
Now, let's look at the exponent 412. We can write it as:
412 = 4(100) + 1(10) + 2
So we need to find the remainder when (432)412 is divided by 10, which is the same as finding the remainder when (432)4(100) * (432)1(10) * (432)2 is divided by 10.
Using the cycles of 2^n unit digits, we know that the unit digit of (432)4(100) is 6, and the unit digit of (432)1(10) is 2.
Now we just need to find the unit digit of (432)2.
(432)2 = 186624
The unit digit is 4.
So the remainder when (432)412 is divided by 10 is the same as the remainder when 6 * 2 * 4 is divided by 10.
6 * 2 * 4 = 48
The unit digit of 48 is 8.
Therefore, the unit digit of (432)412 is 8.
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Community Answer
Find the unit digit of (432)412 × (499)431a)1b)3c)2d)5e)4Correct...
Given:
(432)412 × (499)431
Concept:
9even no. = unit digit 1
9odd no. = unit digit 9
Calculation:
(432)412 × (499)431
Taking unit digits
⇒ 2412 × 9431
As we know unit digit of
21 = 2, 22 = 4, 23 = 8, 24 = 6
⇒ 24(103) × 9431
⇒ 6 × 9
⇒ 54
∴ The unit digit of (432)412 × (499)431 is 4.
(432)412 × (499)431
Concept:
9even no. = unit digit 1
9odd no. = unit digit 9
Calculation:
(432)412 × (499)431
Taking unit digits
⇒ 2412 × 9431
As we know unit digit of
21 = 2, 22 = 4, 23 = 8, 24 = 6
⇒ 24(103) × 9431
⇒ 6 × 9
⇒ 54
∴ The unit digit of (432)412 × (499)431 is 4.
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Find the unit digit of (432)412 × (499)431a)1b)3c)2d)5e)4Correct answer is option 'E'. Can you explain this answer? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about Find the unit digit of (432)412 × (499)431a)1b)3c)2d)5e)4Correct answer is option 'E'. Can you explain this answer? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the unit digit of (432)412 × (499)431a)1b)3c)2d)5e)4Correct answer is option 'E'. Can you explain this answer?.
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