The number of integers between 1 and 500 (both inclusive) that are divisible by 3 or 5 or 7 is ______.Note:This questions appeared as Numerical Answer Type.a)269b)270c)271d)272Correct answer is option 'C'. Can you explain this answer?

Question

Accepted Answer

The general formula for the union of 3 sets is:
(A union B union C) = A + B + C - (A intersect B) - (A intersect C) - (B intersect C) + (A intersect B intersect C).
Assuming,
A = 3, B = 5, C = 7
= 500/3 + 500/5 + 500/7 - 500/3*5 - 500/5*7 - 500/7*3 + 500/105
= 271
Therefore, option C is correct.
Alternate Solution:

Alternate Solution:

Alternate solution
Let a = number divisible by 3
b = number divisible by 5
c = number divisible by 7
n(a) = 166
n(b) = 100
n(c) = 71
n(a∩b) = number divisible by 15 = 33
n(b∩c) = number divisible by 35 = 14
n(a∩c) = number divisible by 21 = 23
n(a∩b∩c) = number divisible by 105 = 4
n(a∪b∪c) = n(a) + n(b) + n(c) - n(a∩b) - n(b∩c) - n(a∩c) + n(a∩b∩c) = 166 + 100 + 71 - 33 - 14 - 23 + 4 = 271

Answer

The general formula for the union of 3 sets is: (A union B union C) = A + B + C – (A intersect B) – (A intersect C) – (B intersect C) + (A intersect B intersect C).
Assuming, A = 3, B = 5, C = 7 = 500/3 + 500/5 + 500/7 – 500/3*5 – 500/5*7 – 500/7*3 + 500/105 = 271 Therefore, option C is correct

Answer

To find the number of integers between 1 and 500 (both inclusive) that are divisible by 3 or 5 or 7, we can use the principle of inclusion-exclusion.

Principle of Inclusion-Exclusion:
The principle of inclusion-exclusion is a counting technique used to find the number of elements in the union of multiple sets.

Step 1: Find the number of integers divisible by 3.
- The first integer divisible by 3 is 3 itself.
- The last integer divisible by 3 in the given range is 498 (500 - (500 mod 3)).
- The number of integers divisible by 3 between 1 and 500 is (498 - 3)/3 + 1 = 166.

Step 2: Find the number of integers divisible by 5.
- The first integer divisible by 5 is 5.
- The last integer divisible by 5 in the given range is 500.
- The number of integers divisible by 5 between 1 and 500 is (500 - 5)/5 + 1 = 100.

Step 3: Find the number of integers divisible by 7.
- The first integer divisible by 7 is 7.
- The last integer divisible by 7 in the given range is 497 (500 - (500 mod 7)).
- The number of integers divisible by 7 between 1 and 500 is (497 - 7)/7 + 1 = 71.

Step 4: Find the number of integers divisible by both 3 and 5.
- The first integer divisible by both 3 and 5 is 15.
- The last integer divisible by both 3 and 5 in the given range is 495 (500 - (500 mod 15)).
- The number of integers divisible by both 3 and 5 between 1 and 500 is (495 - 15)/15 + 1 = 33.

Step 5: Find the number of integers divisible by both 3 and 7.
- The first integer divisible by both 3 and 7 is 21.
- The last integer divisible by both 3 and 7 in the given range is 497 (500 - (500 mod 21)).
- The number of integers divisible by both 3 and 7 between 1 and 500 is (497 - 21)/21 + 1 = 23.

Step 6: Find the number of integers divisible by both 5 and 7.
- The first integer divisible by both 5 and 7 is 35.
- The last integer divisible by both 5 and 7 in the given range is 490 (500 - (500 mod 35)).
- The number of integers divisible by both 5 and 7 between 1 and 500 is (490 - 35)/35 + 1 = 14.

Step 7: Find the number of integers divisible by 3, 5, and 7.
- The first integer divisible by 3, 5, and 7 is 105.
- The last integer divisible by 3, 5, and 7 in the given range is 490 (500 - (500 mod 105)).
- The number of integers divisible by 3, 5, and 7 between 1 and 500 is (490 - 105)/105 + 1 = 4

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