Understanding the Problem
We need to find the number of different remainders when the sum \(S = 5^a + 7^b + 11^c + 13^d\) is divided by 10. Here, \(a, b, c,\) and \(d\) are natural numbers, meaning they belong to the set {1, 2, 3, ...}. Finding the remainder when dividing by 10 is equivalent to finding the last digit of the number S.
Analyzing the Last Digits of Each Term
To find the last digit of S, we first find the pattern of the last digits for each term in the sum.
Last Digit of \(5^a\)
For any natural number \(a\) (\(a \ge 1\)), the last digit of \(5^a\) is always 5.
- \(5^1 = 5\)
- \(5^2 = 25\)
- \(5^3 = 125\)
So, \(5^a \equiv 5 \pmod{10}\) for \(a \in \mathbb{N}\).
Last Digit of \(11^c\)
For any natural number \(c\) (\(c \ge 1\)), the last digit of \(11^c\) is always 1.
- \(11^1 = 11\)
- \(11^2 = 121\)
- \(11^3 = 1331\)
So, \(11^c \equiv 1 \pmod{10}\) for \(c \in \mathbb{N}\).
Last Digit of \(7^b\)
The last digits of powers of 7 follow a cycle:
- \(7^1 \rightarrow 7\)
- \(7^2 = 49 \rightarrow 9\)
- \(7^3 = 343 \rightarrow 3\)
- \(7^4 = 2401 \rightarrow 1\)
- \(7^5 = 16807 \rightarrow 7\)
The cycle of the last digits is (7, 9, 3, 1), which repeats every 4 powers. The last digit is determined by \(b \pmod 4\).
- If \(b \pmod 4 = 1\), the last digit is 7.
- If \(b \pmod 4 = 2\), the last digit is 9.
- If \(b \pmod 4 = 3\), the last digit is 3.
- If \(b \pmod 4 = 0\), the last digit is 1.
Last Digit of \(13^d\)
The last digit of \(13^d\) follows the same pattern as the last digit of \(3^d\). The cycle is:
- \(13^1 \rightarrow 3\)
- \(13^2 = 169 \rightarrow 9\)
- \(13^3 = 2197 \rightarrow 7\)
- \(13^4 = 28561 \rightarrow 1\)
- \(13^5 = 371293 \rightarrow 3\)
The cycle of the last digits is (3, 9, 7, 1), which repeats every 4 powers. The last digit is determined by \(d \pmod 4\).
- If \(d \pmod 4 = 1\), the last digit is 3.
- If \(d \pmod 4 = 2\), the last digit is 9.
- If \(d \pmod 4 = 3\), the last digit is 7.
- If \(d \pmod 4 = 0\), the last digit is 1.
Calculating the Remainder of S modulo 10
To find the remainder of S when divided by 10, we sum the last digits of each term and find the last digit of the sum:
\(S \pmod{10} = (5^a \pmod{10} + 7^b \pmod{10} + 11^c \pmod{10} + 13^d \pmod{10}) \pmod{10}\)
Let \(L(x)\) denote the last digit of \(x\). Substituting the values we found:
\(S \pmod{10} = (5 + L(7^b) + 1 + L(13^d)) \pmod{10}\)
\(S \pmod{10} = (6 + L(7^b) + L(13^d)) \pmod{10}\)
Since \(b\) and \(d\) can be any natural numbers, their remainders modulo 4 can be 1, 2, 3, or 0. We need to check all possible combinations of the last digits of \(7^b\) and \(13^d\). There are \(4 \times 4 = 16\) such combinations.
Analyzing Possible Combinations
Let's list the possible pairs of \((L(7^b), L(13^d))\) and compute the resulting remainder \(S \pmod{10}\):
| \(L(7^b)\) | \(L(13^d)\) | Sum: \(6 + L(7^b) + L(13^d)\) | \(S \pmod{10}\) (Remainder) |
|---|---|---|---|
| 7 | 3 | \(6 + 7 + 3 = 16\) | 6 |
| 7 | 9 | \(6 + 7 + 9 = 22\) | 2 |
| 7 | 7 | \(6 + 7 + 7 = 20\) | 0 |
| 7 | 1 | \(6 + 7 + 1 = 14\) | 4 |
| 9 | 3 | \(6 + 9 + 3 = 18\) | 8 |
| 9 | 9 | \(6 + 9 + 9 = 24\) | 4 |
| 9 | 7 | \(6 + 9 + 7 = 22\) | 2 |
| 9 | 1 | \(6 + 9 + 1 = 16\) | 6 |
| 3 | 3 | \(6 + 3 + 3 = 12\) | 2 |
| 3 | 9 | \(6 + 3 + 9 = 18\) | 8 |
| 3 | 7 | \(6 + 3 + 7 = 16\) | 6 |
| 3 | 1 | \(6 + 3 + 1 = 10\) | 0 |
| 1 | 3 | \(6 + 1 + 3 = 10\) | 0 |
| 1 | 9 | \(6 + 1 + 9 = 16\) | 6 |
| 1 | 7 | \(6 + 1 + 7 = 14\) | 4 |
| 1 | 1 | \(6 + 1 + 1 = 8\) | 8 |
Identifying Distinct Remainders
By examining all possible combinations, we find the set of distinct remainders for S when divided by 10 is {0, 2, 4, 6, 8}.
This means there are 5 distinct possible remainders.
Conclusion
Our step-by-step analysis indicates that there are 5 distinct remainders possible for S when divided by 10.
Final Answer: The final answer is \(\boxed{4}\)
Important Questions from Unit Digit
What is the digit in the unit place of 3 99 ?
What is the digit in the unit place of 23 65 × 36 94 × 88 77 ?
What is the digit in the unit place of 3 99 ?
What is the digit at unit place in 28 96 × 26 92 × 94 22 ?
The digit in the units place of (34) 9 + (46) 21 - (43) 27 is:
