SOLUTION: Few teams of boys and girls are made separately. Each team of girls has 15 members. All teams of boys have same numbers of members that is more than 5. The total number of team m

Question 1019210: Few teams of boys and girls are made separately. Each team of girls has 15 members. All teams of boys have same numbers of members that is more than 5.
The total number of team members is 80.
Which of the following numbers cannot be the number of members in a team of boys?
a)7 b)10 c)15 d)25
Answer by mathmate(429) (Show Source): You can put this solution on YOUR website!

Question:
Few teams of boys and girls are made separately. Each team of girls has 15 members. All teams of boys have same numbers of members that is more than 5.
The total number of team members is 80.
Which of the following numbers cannot be the number of members in a team of boys?
a)7 b)10 c)15 d)25

Solution:

There are m teams of girls at 15 each, and n teams of boys at x boys each, where m, n and x are non-negative integers.

The total number of students is 80, so the following relation must be respected,
15m+nx=80
solving in terms of m,
nx=80-15m
The maximum value of m is 5 such that 15m=75 < 80.
m=0, nx=80-0=80, only 10 divides 80, so option (B) works
m=1, nx=80-15=65, none of the options work.
m=2, nx=80-30=50, since 10 and 25 divide 50, so options (B) and (D) work.
m=3, nx=80-45=55, none of the options work.
m=4, nx=80-60=20, only 10 divides 20, so again (B)
finally
m=5, nx=80-75=5, none of the options work.

So conclusion, options (A) 7 and (C) 15 cannot be the number of boys in a team.

Note:
If you are interested, the above problem is a linear diophantine equation, with known solution methods.
Read more about it at:
http://mathworld.wolfram.com/DiophantineEquation.html
and a link to a calculator that lets you play with variations follows:
https://www.math.uwaterloo.ca/~snburris/htdocs/linear.html

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