Question

In a garden of flowers, a swarm of bees is setting in equal number on flowers. When they settle on two flowers, one bee will be left out. When they settle on three flowers, two bees will be left out. When they settle on four flowers, three bees will be left out. Similarly, when they settle on five flowers no bee will be left out. If there are at most fifty bees, how many bees are there in the swarm?​

Answer 1

The number of bees is 35.

Step-by-step explanation:

Let the number of bees be a

Then

Case: 1

Settle on 2 flowers.

a = 2 b + 1  -----(1)

For some natural number be b

Case 2: a = 3c + 2  -----(2)

Case 3:  a = 4d + 3  ------(3)

Case 4: a = 5e    -----(4)

a < 5 0 ------(5)

From equation 1, 2 and 3.

a + 1 = 2 ( b + 1)

a + 1 = 3 ( c + 1)

a + 1 = 4 ( d + 1)

(a+1) is L.C.M of all R.H.S

(a+1) = 4 x 3 x m   ------(6)

For m = L.C.M  ( b + 1), ( c + 1) , (d+1)

From equation 4 and 6.

a + 1 = 12 m

a = 5 e

a = 12 m - 1

Such that a is a multiplier of 5.

a < 50

e < 10

Now by trial and error.

5 x e = 5 x 7 = 35

12 x 3 - 1 = 36 - 1 = 35

5 e = 12 m - 1 =

e = 7 and  m = 3

The number of bees is 35.