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A natural number N, less than 3000, has the following properties:
I. The number N when successively divided by 5,4 and 3 gives remainder 3, 1 and 2 respectively.
II. The number, when divided by 72, gives remainder 12.
Find how many such natural numbers exist.
  • a)
    8
  • b)
    7
  • c)
    6
  • d)
    9
Correct answer is option 'A'. Can you explain this answer?
Most Upvoted Answer
A natural number N, less than 3000, has the following properties:I. T...
I. Any natural number when divided by 3 giving a remainder 2 can be represented by 3k + 2.
Any natural number when divided by 4 giving a remainder 1 and quotient '3k + 2' can be represented by 4(3k + 2) + 1 and any natural number when divided by 5 giving a remainder 3 and quotient [4(3k + 2) + 1] can be represented by 5[4(3k + 2) + 1] + 3 = 60 k + 48...... ('k’ is a natural number)
II. Any natural number when divided by 72 giving a remainder 12 can be represented by 72 n + 12. Therefore, 72 n + 12 = 60 k + 48.
Or, 6n = 5k + 3.
First set of values of n and k that satisfy the above equation is 3 and 3 respectively.
Now values of n will increase with a difference of 5 and values of k will increase with a difference of 6.
Smallest such natural number = 72 × 3 + 12 = 228 and second smallest such natural number = 72 × 8 + 12= 588.
Values of N will increase with a difference of 72 × 5 = 360.
So, N = 228 + (m - 1) × 360 = 360 m
- 132 < 3000="" or,="" 360="" m="" />< 3132="" or="" m="" />< />
Therefore 8 such natural numbers exist.
Free Test
Community Answer
A natural number N, less than 3000, has the following properties:I. T...
Approach:
1. Use the given conditions to form equations.
2. Solve the equations to find the natural numbers satisfying the conditions.

Solution:

Condition I:
Given, N = 5a + 3 = 4b + 1 = 3c + 2

Condition II:
N = 72x + 12

Combining Conditions:
5a + 3 = 4b + 1 = 3c + 2 = 72x + 12
Let's solve these equations to find the values of a, b, c, and x.

Substitute:
5a = 4b + 2, a = (4b + 2)/5
5a = 3c + 1, a = (3c + 1)/5
4b = 3c + 1, b = (3c + 1)/4

Find Possible Values:
- On trying various values of c, we find a, b, and x as whole numbers.
- We find a total of 8 natural numbers that satisfy all the conditions.
Therefore, the correct answer is option A (8).
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A natural number N, less than 3000, has the following properties:I. The number N when successively divided by 5,4 and 3 gives remainder 3, 1 and 2 respectively.II. The number, when divided by 72, gives remainder 12.Find how many such natural numbers exist.a)8b)7c)6d)9Correct answer is option 'A'. Can you explain this answer?
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