What is the unit digit in the multiplication of 1 × 3 × 5 × 7 × 9 × ... × 999?
5
Finding the Unit Digit of a Product
The question asks for the unit digit of the product of all odd numbers from 1 to 999. The product is given by:
\(1 \times 3 \times 5 \times 7 \times 9 \times ... \times 999\)
To find the unit digit of a large product, we only need to focus on the unit digits of the numbers being multiplied.
The numbers in the product are all the odd numbers from 1 up to 999. Let's look at some of the numbers and their unit digits:
- 1 (unit digit is 1)
- 3 (unit digit is 3)
- 5 (unit digit is 5)
- 7 (unit digit is 7)
- 9 (unit digit is 9)
- 11 (unit digit is 1)
- 13 (unit digit is 3)
- 15 (unit digit is 5)
- ... and so on.
The sequence of unit digits of odd numbers repeats as 1, 3, 5, 7, 9, 1, 3, 5, 7, 9, ...
In the given product \(1 \times 3 \times 5 \times 7 \times 9 \times ... \times 999\), one of the numbers being multiplied is 5. All other numbers in the product (1, 3, 7, 9, 11, 13, 15, 17, 19, ..., 999) are odd numbers.
Let's consider the effect of multiplying a number ending in 5 by other numbers:
- If a number ending in 5 is multiplied by any odd number, the unit digit of the result will be 5. For example:
- \(5 \times 1 = 5\) (unit digit 5)
- \(5 \times 3 = 15\) (unit digit 5)
- \(5 \times 7 = 35\) (unit digit 5)
- \(5 \times 9 = 45\) (unit digit 5)
- \(5 \times 13 = 65\) (unit digit 5)
- If a number ending in 5 is multiplied by any even number, the unit digit of the result will be 0. For example:
- \(5 \times 2 = 10\) (unit digit 0)
- \(5 \times 4 = 20\) (unit digit 0)
- \(5 \times 6 = 30\) (unit digit 0)
- \(5 \times 8 = 40\) (unit digit 0)
In our product \(1 \times 3 \times 5 \times 7 \times 9 \times ... \times 999\), we are multiplying 5 by a series of other odd numbers (1, 3, 7, 9, 11, 13, ..., 999). Since 5 is being multiplied exclusively by odd numbers (and itself, which is odd), the unit digit of the final product must be 5.
Thus, the unit digit in the multiplication of \(1 \times 3 \times 5 \times 7 \times 9 \times ... \times 999\) is 5.
Revision Table: Understanding Unit Digits in Products
| Numbers in Product | Contains 5? | Contains any Even Number? | Unit Digit of Product |
|---|---|---|---|
| Odd numbers only (and includes 5) | Yes | No | 5 |
| Any numbers (and includes 5 and at least one even number) | Yes | Yes | 0 |
| Any numbers (but does not include 5) | No | May be or May not be | Depends on the unit digits of other numbers |
Additional Information: Unit Digit Cycles
The unit digits of powers of a number often follow a repeating pattern or cycle. For example:
- Powers of 1: \(1^1=1, 1^2=1, 1^3=1, ...\) Cycle: 1 (length 1)
- Powers of 2: \(2^1=2, 2^2=4, 2^3=8, 2^4=16 (\text{unit digit 6}), 2^5=32 (\text{unit digit 2}), ...\) Cycle: 2, 4, 8, 6 (length 4)
- Powers of 3: \(3^1=3, 3^2=9, 3^3=27 (\text{unit digit 7}), 3^4=81 (\text{unit digit 1}), 3^5=243 (\text{unit digit 3}), ...\) Cycle: 3, 9, 7, 1 (length 4)
- Powers of 5: \(5^1=5, 5^2=25 (\text{unit digit 5}), 5^3=125 (\text{unit digit 5}), ...\) Cycle: 5 (length 1)
- Powers of 10: \(10^1=10 (\text{unit digit 0}), 10^2=100 (\text{unit digit 0}), ...\) Cycle: 0 (length 1)
Understanding these cycles can help in finding the unit digit of large powers. For products, identifying factors with unit digits 0 or 5 is crucial as they often determine the final unit digit quickly.
Similar Questions
Which one of the following statements best reflects the critical message conveyed by the author of the passage?
With reference to the above passage, the following assumptions have been made:
I. No country needs to depend on ecosystems to boost national income.
II. Resource-rich countries need to share their resources with those of scant resources so as to prevent the degradation of ecosystems.
Which of the above assumptions is/are valid?Which one of the following statements best reflects the central idea of the passage?
With reference to the above passage, the following assumptions have been made:
I. Path-dependent green investments will eventually most likely benefit growth as well as public finances in a country like India.
II. If other green technologies follow the same pattern as that of solar energy, there will most likely be an easy green transition.
Which of the above assumptions is/are valid?Three prime numbers p, q and r, each less than 20, are such that p − q = q − r. How many distinct possible values can we get for (p + q + r)?
How many possible values of (p + q + r) are there satisfying 1/p + 1/q + 1/r = 1, where p, q and r are natural numbers (not necessarily distinct)?
What comes at X and Y respectively in the following sequence?
January, January, December, October, X, March, October, Y, SeptemberTeam X scored a total of N runs in 20 overs. Team Y tied the score in 10% less overs. Had Team Y’s average run rate (runs per over) been 50% higher, the scores would have been tied in 12 overs. How many runs were scored by Team X?
The price (p) of a commodity is first increased by k%; then decreased by k%; again increased by k%; and again decreased by k%. If the new price is q, then what is the relation between p and q?
Which one of the following statements best reflects the most logical, rational and pragmatic message conveyed by the author of the passage?
Important Questions from Miscellaneous Topics
Which one of the following statements best reflects the critical message conveyed by the author of the passage?
With reference to the above passage, the following assumptions have been made:
I. No country needs to depend on ecosystems to boost national income.
II. Resource-rich countries need to share their resources with those of scant resources so as to prevent the degradation of ecosystems.
Which of the above assumptions is/are valid?Which one of the following statements best reflects the central idea of the passage?
With reference to the above passage, the following assumptions have been made:
I. Path-dependent green investments will eventually most likely benefit growth as well as public finances in a country like India.
II. If other green technologies follow the same pattern as that of solar energy, there will most likely be an easy green transition.
Which of the above assumptions is/are valid?Three prime numbers p, q and r, each less than 20, are such that p − q = q − r. How many distinct possible values can we get for (p + q + r)?
