Question
prove that 140 x 141 x 142 x ...x 159 x 160 is divisible by 4321
Answers
GPT-4o mini
To determine whether the product \( 140 \times 141 \times 142 \times \ldots \times 159 \times 160 \) is divisible by \( 4321 \), we first need to factor \( 4321 \).
### Step 1: Factoring 4321
To factor \( 4321 \), we can begin by testing for divisibility by prime numbers.
1. **Check divisibility by 2**: \( 4321 \) is odd, so it is not divisible by \( 2 \).
2. **Check divisibility by 3**: The sum of the digits \( 4 + 3 + 2 + 1 = 10 \) is not divisible by \( 3 \).
3. **Check divisibility by 5**: \( 4321 \) does not end in \( 0 \) or \( 5 \), so it is not divisible by \( 5 \).
4. **Check divisibility by 7**: \( 4321 \div 7 \approx 617.2857 \). Not divisible.
5. **Check divisibility by 11**: \( 4321 \div 11 \approx 393.727 \). Not divisible.
6. **Check divisibility by 13**: \( 4321 \div 13 = 333.1538 \). Not divisible.
7. **Check divisibility by 17**: \( 4321 \div 17 = 254.1765 \). Not divisible.
8. **Check divisibility by 19**: \( 4321 \div 19 = 227.421 \). Not divisible.
9. **Check divisibility by 23**: \( 4321 \div 23 = 188.3043 \). Not divisible.
10. **Check divisibility by 29**: \( 4321 \div 29 = 149.6897 \). Not divisible.
11. **Check divisibility by 37**: \( 4321 \div 37 = 116.027 \). Not divisible.
12. **Check divisibility by 41**: \( 4321 \div 41 = 105.9024 \). Not divisible.
13. **Check divisibility by 43**: \( 4321 \div 43 = 100.4884 \). Not divisible.
14. **Check divisibility by 47**: \( 4321 \div 47 = 91.936 \). Not divisible.
15. **Check divisibility by 53**: \( 4321 \div 53 = 81.1321 \). Not divisible.
16. **Check divisibility by 59**: \( 4321 \div 59 = 73.1525 \). Not divisible.
17. **Check divisibility by 61**: \( 4321 \div 61 \approx 70.0164 \). Not divisible.
18. **Check divisibility by 67**: \( 4321 \div 67 \approx 64.4627 \). Not divisible.
19. **Check divisibility by 71**: \( 4321 \div 71 \approx 60.0141 \). Not divisible.
20. **Check divisibility by 73**: \( 4321 \div 73 = 59.0521 \). Not divisible.
21. **Check divisibility by 79**: \( 4321 \div 79 = 54.1722 \). Not divisible.
22. **Check divisibility by 83**: \( 4321 \div 83 = 52.1529 \). Not divisible.
23. **Check divisibility by 89**: \( 4321 \div 89 \approx 48.20 \). Not divisible.
24. **Check divisibility by 97**: \( 4321 \div 97 = 44.585 \). Not divisible.
25. **Check divisibility by 103**: \( 4321 \div 103 = 41.9305 \). Not divisible.
26. **Check divisibility by 109**: \( 4321 \div 109 = 39.6541 \). Not divisible.
27. **Check divisibility by 113**: \( 4321 \div 113 = 38.3654 \). Not divisible.
28. - Finally, check **121**: \( 4321 = 121 \times 37 \).
Thus, the factorization is \( 4321 = 37 \times 117 \).
### Step 2: Verify divisibility by 37 and 113
Next, we verify whether the product \( 140 \times 141 \times \ldots \times 160 \) contains these factors.
1. **Number of terms**: \( 140, 141, 142, \ldots, 160 \) has \( 160 - 140 + 1 = 21 \) terms ranging from \( 140 \) to \( 160 \).
2. **Divisibility by 37**:
- We need to find at least one term divisible by \( 37 \).
- The multiples of \( 37 \) within our range are \( 148 \) (i.e., \( 37 \times 4 = 148 \)).
- Thus, \( 148 \) is a term in the product and contributes the factor \( 37 \).
3. **Divisibility by 113**:
- Similarly, check if there's a term divisible by \( 113 \).
- The only multiple of \( 113 \) within the range goes above \( 160 \) and does not exist in our terms.
- Therefore, we can only rely on factor \( 37 \).
### Conclusion
Since \( 4321 = 37 \times 117 \), and based on our computation, while \( 37 \) appears in the product, \( 117 \) does not exist in that range \( 140 \) to \( 160 \).
Thus, the product \( 140 \times 141 \times \ldots \times 160 \) **is not divisible** by \( 4321 \) as it's not divisible by both primes in \( 117 \).
As such, \( 4321 \) is not fully represented in the product of numbers from \( 140 \) to \( 160 \) completely; although \( 37 \) is present but may not fulfill multiplicity terms to represent \( 4321 \).
### Step 1: Factoring 4321
To factor \( 4321 \), we can begin by testing for divisibility by prime numbers.
1. **Check divisibility by 2**: \( 4321 \) is odd, so it is not divisible by \( 2 \).
2. **Check divisibility by 3**: The sum of the digits \( 4 + 3 + 2 + 1 = 10 \) is not divisible by \( 3 \).
3. **Check divisibility by 5**: \( 4321 \) does not end in \( 0 \) or \( 5 \), so it is not divisible by \( 5 \).
4. **Check divisibility by 7**: \( 4321 \div 7 \approx 617.2857 \). Not divisible.
5. **Check divisibility by 11**: \( 4321 \div 11 \approx 393.727 \). Not divisible.
6. **Check divisibility by 13**: \( 4321 \div 13 = 333.1538 \). Not divisible.
7. **Check divisibility by 17**: \( 4321 \div 17 = 254.1765 \). Not divisible.
8. **Check divisibility by 19**: \( 4321 \div 19 = 227.421 \). Not divisible.
9. **Check divisibility by 23**: \( 4321 \div 23 = 188.3043 \). Not divisible.
10. **Check divisibility by 29**: \( 4321 \div 29 = 149.6897 \). Not divisible.
11. **Check divisibility by 37**: \( 4321 \div 37 = 116.027 \). Not divisible.
12. **Check divisibility by 41**: \( 4321 \div 41 = 105.9024 \). Not divisible.
13. **Check divisibility by 43**: \( 4321 \div 43 = 100.4884 \). Not divisible.
14. **Check divisibility by 47**: \( 4321 \div 47 = 91.936 \). Not divisible.
15. **Check divisibility by 53**: \( 4321 \div 53 = 81.1321 \). Not divisible.
16. **Check divisibility by 59**: \( 4321 \div 59 = 73.1525 \). Not divisible.
17. **Check divisibility by 61**: \( 4321 \div 61 \approx 70.0164 \). Not divisible.
18. **Check divisibility by 67**: \( 4321 \div 67 \approx 64.4627 \). Not divisible.
19. **Check divisibility by 71**: \( 4321 \div 71 \approx 60.0141 \). Not divisible.
20. **Check divisibility by 73**: \( 4321 \div 73 = 59.0521 \). Not divisible.
21. **Check divisibility by 79**: \( 4321 \div 79 = 54.1722 \). Not divisible.
22. **Check divisibility by 83**: \( 4321 \div 83 = 52.1529 \). Not divisible.
23. **Check divisibility by 89**: \( 4321 \div 89 \approx 48.20 \). Not divisible.
24. **Check divisibility by 97**: \( 4321 \div 97 = 44.585 \). Not divisible.
25. **Check divisibility by 103**: \( 4321 \div 103 = 41.9305 \). Not divisible.
26. **Check divisibility by 109**: \( 4321 \div 109 = 39.6541 \). Not divisible.
27. **Check divisibility by 113**: \( 4321 \div 113 = 38.3654 \). Not divisible.
28. - Finally, check **121**: \( 4321 = 121 \times 37 \).
Thus, the factorization is \( 4321 = 37 \times 117 \).
### Step 2: Verify divisibility by 37 and 113
Next, we verify whether the product \( 140 \times 141 \times \ldots \times 160 \) contains these factors.
1. **Number of terms**: \( 140, 141, 142, \ldots, 160 \) has \( 160 - 140 + 1 = 21 \) terms ranging from \( 140 \) to \( 160 \).
2. **Divisibility by 37**:
- We need to find at least one term divisible by \( 37 \).
- The multiples of \( 37 \) within our range are \( 148 \) (i.e., \( 37 \times 4 = 148 \)).
- Thus, \( 148 \) is a term in the product and contributes the factor \( 37 \).
3. **Divisibility by 113**:
- Similarly, check if there's a term divisible by \( 113 \).
- The only multiple of \( 113 \) within the range goes above \( 160 \) and does not exist in our terms.
- Therefore, we can only rely on factor \( 37 \).
### Conclusion
Since \( 4321 = 37 \times 117 \), and based on our computation, while \( 37 \) appears in the product, \( 117 \) does not exist in that range \( 140 \) to \( 160 \).
Thus, the product \( 140 \times 141 \times \ldots \times 160 \) **is not divisible** by \( 4321 \) as it's not divisible by both primes in \( 117 \).
As such, \( 4321 \) is not fully represented in the product of numbers from \( 140 \) to \( 160 \) completely; although \( 37 \) is present but may not fulfill multiplicity terms to represent \( 4321 \).