both questions please 8 If eggs are removed from a basket 2

both questions please

8. If eggs are removed from a basket 2, 3, 4, 5 and 6 at a time, there remain, respectively, 1, 2, 3, 4 and 5 eggs. But if eggs are removed 7 at a time, no eggs remain. What is the least number of eggs that could have been in the basket? 9. What is the smallest number of lobsters in a tank if 1 lobster is left when they are removed 2, 3, 5 or 7 at a time, but no lobsters are left over when they removed 11 at a time?

Solution

8 ) Let X be the number of eggs in the basket.

X is 1 less than a multiple of 2, so X=2A-1 for some positive integer A.

similarly X=3B-1 for some positive integer B, X=4C-1 for some positive integer C, X=5D-1 for some positive integer D, X=6E-1 for some positive integer E and X=7F for some positive integer F.

Therefore X= K*LCM(2,3,4,5,6) - 1 ( for some K )

LCM(2,3,4,5,6)= 60

Hence least X possible is for K=1 but then X=60-1=59 which not a multiple of 7.

therefore least possible K is equal to 2. Hence X=2*60-1 = 120-1 =119 (answer)

9) Let X be the number of Lobsters in the tank.

(Using Similar approach as in question 8)

LCM(2,3,5,7) =210

therefore X= K*LCM(2,3,5,7) -1

X=K*210-1 and also X is a multiple of 11.

Least K possible =1

therefore X=210-1 = 209 (Answer)