(a) Total number of positive integral solution of the
X X X = 60 is 18k, then k is equal to
Answers
Given : X₁*X₂*X₃ = 60
Total number of positive integral solution is 18k
To Find : value of k
Solution:
if ordered pairs are considers
X₁*X₂*X₃ = 60
1 * 1 * 60 = 60
1 * 2 * 30 = 60
1 * 3 * 20 = 60
1 * 4 * 15 = 60
1 * 5 * 12 = 60
1 * 6 * 10 = 60
1 * 10 * 6 = 60
1 * 12 * 5 = 60
1 * 15 * 4 = 60
1 * 20 * 3 = 60
1 * 30 * 2 = 60
1 * 60 * 1 = 60
2 * 1 * 30 = 60
2 * 2 * 15 = 60
2 * 3 * 10 = 60
2 * 5 * 6 = 60
2 * 6 * 5 = 60
2 * 10 * 3 = 60
2 * 15 * 2 = 60
2 * 30 * 1 = 60
3 * 1 * 20 = 60
3 * 2 * 10 = 60
3 * 4 * 5 = 60
3 * 5 * 4 = 60
3 * 10 * 2 = 60
3 * 20 * 1 = 60
4 * 1 * 15 = 60
4 * 3 * 5 = 60
4 * 5 * 3 = 60
4 * 15 * 1= 60
5 * 1 * 12 = 60
5 * 2 * 6 = 60
5 * 3 * 4 = 60
5 * 4 * 3 = 60
5 * 6 * 2 = 60
5 * 12 * 1 = 60
6 * 1* 10 = 60
6 * 2 * 5 = 60
6 * 5 * 2 = 60
6 * 10 * 1 = 60
10 * 1 * 6 = 60
10 * 2 * 3= 60
10 * 3 * 2 = 60
10 * 1 * 6 = 60
12 * 1 * 5 = 60
12 * 5 * 1 = 60
15 * 1 * 4 = 60
15 * 2 * 2 = 60
15 * 4 * 1 = 60
20 * 1 * 3 = 60
20 * 3 * 1= 60
30 * 1 * 2 = 60
30 * 2 * 1 = 60
60 * 1 * 1 = 60
Then 52 solutions
18k = 54
=> k = 54/18 = 3
Hence value of k = 3
if ordered pair not considered
X₁*X₂*X₃ = 60
1 * 1 * 60 = 60
1 * 2 * 30 = 60
1 * 3 * 20 = 60
1 * 4 * 15 = 60
1 * 5 * 12 = 60
1 * 6 * 10 = 60
2 * 2 * 15 = 60
2 * 3 * 10 = 60
2 * 5 * 6 = 60
3 * 4 * 5 = 60
18k = 10 =>k = 5/9
if X₁,X₂,X₃ are unique digits and not order pair
X₁*X₂*X₃ = 60
1 * 2 * 30 = 60
1 * 3 * 20 = 60
1 * 4 * 15 = 60
1 * 5 * 12 = 60
1 * 6 * 10 = 60
2 * 3 * 10 = 60
2 * 5 * 6 = 60
3 * 4 * 5 = 60
18k = 8 =>k = 4/9
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