Problema Solution
Dina has a bag of beads to make a necklace. When asked to tell how many beads are in the bag, she said, " I can make sets of 2 beads, 4 beads, and 5 beads and not have any left. When I divide the beads into 9 sets, I have 7 beads left over. What is the least number of beads Dina can have?
Answer provided by our tutors
let n = the number of beads
I can make sets of 2 beads, 4 beads, and 5 beads and not have any left means
2|n, 4|n and 5|n follows LCM(2, 4, 5) = 20|n and n>= 20
When I divide the beads into 9 sets, I have 7 beads left over
n = 9k + 7, where k is integer
since n>=20 follows k>=2
9k + 7 will end on 0 if 9k ends on 3 and that is possible for k ending on 7
for k = 7
n = 9*7 + 7 = 70 but 70 is not dividable by 20
for k = 17 we have
n = 9*17 + 7 = 160 and 20|160
the solution is 160
indeed 2|160, 4|160 and 5|160
160 divided by 9 gives a reminder of 7.