Unit Digit Calculation Strategy
To find the unit digit of a product of numbers raised to powers, we only need to consider the unit digits of each number and its power.
- The unit digit of a product is determined by the unit digits of the factors involved.
- We will find the unit digit of each term (\(1^1, 3^3, 5^5, 7^7, 9^9\)) separately.
- Then, we find the unit digit of the product of these unit digits.
Unit Digits of Individual Terms
- Unit digit of \(1^1\): The unit digit of \(1\) raised to any positive integer power is always 1.
Unit digit = 1.
- Unit digit of \(3^3\): \(3^3 = 3 \times 3 \times 3 = 27\).
Unit digit = 7.
- Unit digit of \(5^5\): The unit digit of \(5\) raised to any positive integer power is always 5.
Unit digit = 5.
- Unit digit of \(7^7\): The unit digits of powers of 7 follow a cycle: \(7^1 \rightarrow 7\), \(7^2 \rightarrow 9\), \(7^3 \rightarrow 3\), \(7^4 \rightarrow 1\). The cycle length is 4. To find the unit digit of \(7^7\), we find the remainder of \(7 \div 4\), which is 3. The unit digit corresponds to the 3rd element in the cycle.
Unit digit = 3.
- Unit digit of \(9^9\): The unit digits of powers of 9 follow a cycle: \(9^1 \rightarrow 9\), \(9^2 \rightarrow 1\). The cycle length is 2. To find the unit digit of \(9^9\), we find the remainder of \(9 \div 2\), which is 1. The unit digit corresponds to the 1st element in the cycle.
Unit digit = 9.
Unit Digit of the Final Product
Now, we multiply the unit digits found above: 1, 7, 5, 3, 9.
The unit digit of the expression \(1^1 \times 3^3 \times 5^5 \times 7^7 \times 9^9\) is the unit digit of the product \(1 \times 7 \times 5 \times 3 \times 9\).
We can calculate the product of the unit digits:
- \(1 \times 7 = 7\)
- \(7 \times 5 = 35\). The unit digit is 5.
- Since we have encountered a unit digit of 5, and all the remaining numbers to be multiplied (3 and 9) are odd, the final unit digit will be 5. Any integer multiplied by 5 results in a number ending in 0 or 5. If the other factor is odd, the result ends in 5.
- Continuing the calculation for completeness: Unit digit of \((...5) \times 3\) is 5. Unit digit of \((...5) \times 9\) is 5.
Therefore, the unit digit of the number \(1^1 \times 3^3 \times 5^5 \times 7^7 \times 9^9\) is 5.
Similar Questions
What is the sum of the largest and the smallest 4-digit numbers made by using single digit prime numbers (without repetition)?
- XYZ is a 3-digit number, where X, Y, Z are distinct non-zero digits. The difference between the two 3-digit numbers XYZ and YXZ is 90. How many possible values exist for the sum (X+Y) ?
- The number 199 can be written as \(m^2 - n^2\), where \(m, n\) are natural numbers (\(m > n\)). What is the value of \(mn\)?
- How many numbers of the form \(2^n - 1\) and less than 2000 are prime?
- If \(\frac{1}{a + \frac{1}{b + \frac{1}{c + \frac{1}{d + \frac{1}{e}}}}} = \frac{421}{972}\), then what is the value of \(a \times b \times c \times d \times e\)?
- Let \(p\) and \(q\) be natural numbers such that \(q > p\). What is the largest value of \(p\) such that \(q^2 - 5p-4\) is negative?
- Let \(x\) and \(y\) be natural numbers, each less than 20, such that \(x\), \(y\), \(x+y\) and \(x-y\) are prime numbers. How many such combinations of \((x, y, x+y, x - y)\) are possible?
What is the remainder when
\((17^{25} +19^{25})\)
is divided by 18?
Let \(x = n(n+1)(n+2)\), where \(n\) is an even natural number. Which of the following statements is/are correct?
I. \(x\) is always divisible by 48.
II. \(x^2\) is always divisible by 144.
Select the answer using the code given below.
- Let XYZ be a 3-digit number. Let D be the difference between XYZ and ZYX. What is the remainder when D is divided by 99?
Important Questions from Number System
What is the value of 1 2 + 2 2 + 3 2 + ......21 2 ?
Which sequence is correct to represent the hierarchical chain of number system?
(Where N - Natural Numbers
W - Whole Numbers
Q - Rational Numbers
Z - Integers)
What must be added to 45680 to make it exactly divisible by 9?
How many zeroes are there at the end of the following product?
1 x 5 x 10 x 15 x 20 x 25 x 30 x 35 x 40 x 45 x 50 x 55 x 60
Let XYZ be a three-digit number, where (x + y + Z) is not a multiple of 3. Then (XYZ + YZX + ZXY) is not divisible by
