7. Given that w=7r+6r+5r+4r+3r. Which of the terms listed below may be added to w to the resulting sum will be divisible by 7 for every positive integer r?
A.7r
B.6r
C.5r
D.4r
E.3r
since w = 25r, we need to increase by 3r, making the result 28r, which is always divisible by 7, since 7 divides 28, regardless of r.
w=7r+6r+5r+4r+3r
w = 25r
the next multiple of 7r would be 28r for any positive integer of r
so we need to add 3r
To find the terms that can be added to w to get a sum divisible by 7 for every positive integer r, we need to determine which terms are divisible by 7.
Given that w = 7r + 6r + 5r + 4r + 3r, we can simplify it by combining like terms:
w = (7 + 6 + 5 + 4 + 3)r = 25r
Now, we need to find the terms that can be added to 25r to get a sum divisible by 7.
Let's analyze the options:
A. 7r: This term is already a multiple of 7, so it can be added to 25r.
B. 6r: This term is not necessarily a multiple of 7, as it can be written as 7r - r. Therefore, it cannot be added to 25r.
C. 5r: Similar to option B, this term is not necessarily a multiple of 7, as it can be written as 7r - 2r. Therefore, it cannot be added to 25r.
D. 4r: Similar to options B and C, this term is not necessarily a multiple of 7, as it can be written as 7r - 3r. Therefore, it cannot be added to 25r.
E. 3r: Similar to options B, C, and D, this term is not necessarily a multiple of 7, as it can be written as 7r - 4r. Therefore, it cannot be added to 25r.
Therefore, the only term that can be added to w to get a sum divisible by 7 for every positive integer r is A. 7r.
To determine which term can be added to w to ensure that the sum is divisible by 7 for every positive integer r, we need to consider the divisibility rule for 7.
Divisibility Rule for 7:
A number is divisible by 7 if and only if the difference between twice the units digit and the remaining digits is divisible by 7.
Let's apply this rule to each option:
A. 7r: The units digit is 7, so twice the units digit is 14. Since the sum of the remaining digits is 0, the difference is 14 - 0 = 14. We cannot determine if 14 is divisible by 7 because we do not have enough information. This term does not necessarily contribute to a sum divisible by 7 for every positive integer r.
B. 6r: The units digit is 6, so twice the units digit is 12. Again, there is not enough information to determine if 12 is divisible by 7. This term does not necessarily contribute to a sum divisible by 7 for every positive integer r.
C. 5r: The units digit is 5, so twice the units digit is 10. Similarly, there is not enough information to determine if 10 is divisible by 7. This term does not necessarily contribute to a sum divisible by 7 for every positive integer r.
D. 4r: The units digit is 4, so twice the units digit is 8. Again, there is not enough information to determine if 8 is divisible by 7. This term does not necessarily contribute to a sum divisible by 7 for every positive integer r.
E. 3r: The units digit is 3, so twice the units digit is 6. Unlike the previous options, the difference between 6 and the remaining digits, which is 0 in this case, is always 6. Since 6 is divisible by 7, this term will contribute to a sum divisible by 7 for every positive integer r.
Therefore, the term that can be added to w to ensure a sum divisible by 7 for every positive integer r is 3r (option E).