A farmer has a rose garden. Every day he either plucks 7 or 6 or 24 or 23…
2024
A farmer has a rose garden. Every day he either plucks 7 or 6 or 24 or 23 roses. The rose plants are intelligent and when the farmer plucks these numbers of roses, the next day 37 or 36 or 9 or 18 new roses bloom in the garden respectively. On Monday, he counts 189 roses in the garden. He plucks the roses as per his plan on consecutive days and the new roses bloom as per intelligence of the plants mentioned above. After some days which of the following can be the number of roses in the garden?
- A.
10
- B.
5
- C.
0
- D.
4
Attempted by 426 students.
Show answer & explanation
Correct answer: D
Key idea: every operation changes the total number of roses by a multiple of 5, so the number of roses is invariant modulo 5.
Pluck 7, then 37 grow: net change = +30.
Pluck 6, then 36 grow: net change = +30.
Pluck 24, then 9 grow: net change = -15.
Pluck 23, then 18 grow: net change = -5.
All net changes are multiples of 5, so the number of roses is fixed modulo 5. The starting count 189 is congruent to 4 modulo 5, so only numbers congruent to 4 modulo 5 can occur.
Among the choices, 10, 5, and 0 are congruent to 0 modulo 5 and therefore cannot be reached. 4 is congruent to 4 modulo 5 and is reachable.
Explicit construction: pluck 23 roses for 37 consecutive days. Each day this reduces the total by 5, so after 37 days the total decreases by 5 × 37 = 185 and becomes 189 − 185 = 4.