Answer :
Final Answer:
The least value of [k] is 1: 1: 1. None of the option is correct.
Explanation:
Solve for the values of a, b, and c:
From the equation 5a + 12b + 13c = 52, we can't directly solve for individual values of a, b, and c due to multiple unknowns. However, we can find ratios between them.
Eliminate one variable:
To eliminate one variable and express the others in terms of it, we can choose a common factor based on their coefficients. Since 5 and 13 share a common factor of 1, let's express b and c in terms of a:
12b = 60a (divide both sides by 2)
13c = 52a (divide both sides by 13)
Therefore, b = 5a and c = 4a.
Find the least integer values of a, b, and c:
We need to find the smallest positive integers for a, b, and c that satisfy the equation 5a + 12b + 13c = 52. Substituting b = 5a and c = 4a, we get:
5a + 12(5a) + 13(4a) = 52
65a = 52
a = 4/5 (smallest positive integer solution)
Now, substitute a back to find b and c:
b = 5(4/5) = 4
c = 4(4/5) = 16/5
Calculate the ratios:
Divide each value by their greatest common factor (GCD) which is 4:
a:b:c = 1/5 : 1 : 4/5 = 1 : 5 : 4
Apply the greatest integer function:
Taking the greatest integer of each ratio gives us:
[a]:[b]:[c] = [1/5]:[1]:[4/5] = 0: 1: 0
Simplify:
Since we can't have a ratio of 0 (it implies dividing by 0), we need to increase the first ratio by 1 while maintaining the relative proportionality. Adding 1 to 0/5 keeps the same proportion relative to 1, resulting in:
1: 1: 0 -> 1: 1: 0 + 1 -> 1: 1: 1
Therefore, the least value of [k] is 1: 1: 1. None of the option is correct.
