Question

An NGO was formed to help poor patients admitted to a hospital. They collected fund as contribution and decided to distribute the collection equally as a multiple of 1000 to the patients. They also wanted to keep some amount towards their organisational expenses. When they counted the contribution, it was found that if the amount is to be distributed equally (as multiple of thousand ) among 7 patients Rs.5,000 will be left, if distributed equally to 5 people Rs 4,000 will be left and if distributed equally to 2 people, nothing will be left. If they decide to help 5 people , what is the minimum amount each will receive ?

• A. 9,000
• B. 11,000
• C. 13,000
• D. 15,000

Answer 1

Answer:

A. 9,000

Let the total contribution be ₹N

We are given:

Dividing among 7 patients, ₹5,000 is left:

$N\equiv5000\pmod{7000}$

Dividing among 5 patients, ₹4,000 is left:

$N\equiv4000\pmod{5000}$

Dividing among 2 patients, nothing is left:

$N\equiv0\pmod{2000}$

We need the minimum amount each of 5 people will receive.

Step 1: Use the condition for 2 people

Since N is divisible by 2000, possible amounts are:

$2000,4000,6000,8000,10000,\ldots$

We need: $N\equiv4000\pmod{5000}$

So possible values are:

$4000,9000,14000,19000,24000,29000,34000,\ldots$

Among these, we need a number divisible by 2000.

The first such number satisfying all conditions is:

$\boxed{14000}$

Check: $14000\div7=2000 \quad\text{with }5000\text{ left}$

$14000\div5=2000 \quad\text{with }4000\text{ left}$

$14000\div2=7000 \quad\text{with no remainder}$

Step 2: Find the amount received by 5 people

The ₹4,000 is kept for organisational expenses.

So the amount distributed is:

$14000-4000=10000$

For 5 people:

$\frac{10000}{5}=2000$

Therefore,

Minimum amount each will receive = INR 2,000

Answer: ₹2,000