Total number of numbers lying in the range of 1331 and 3113 which are neither divisible by 2,3 or 5 is A. 477 B. 594 C. 653 D. none of these i couldn't get the answer so. who ever answer it please explain your ans too?

Question

Answer

Total number of numbers lying in the range of 1331 and 3113 which are neither divisible by 2, 3, or 5.

Approach:
To find the total number of numbers in the given range that are not divisible by 2, 3, or 5, we can use the principle of inclusion-exclusion. We need to subtract the numbers divisible by 2, 3, or 5 from the total count of numbers in the given range.

Steps:

1. Find the total count of numbers in the given range.
2. Find the count of numbers divisible by 2, 3, or 5.
3. Subtract the count of divisible numbers from the total count to get the desired result.

Count of Numbers in the Given Range:
The range of numbers lies between 1331 and 3113 (inclusive). To find the count of numbers in this range, we can subtract the lower limit from the upper limit and add 1 (to include both limits).

Count of numbers in the range = 3113 - 1331 + 1 = 1783

Count of Numbers Divisible by 2:
To find the count of numbers divisible by 2 in the given range, we need to find the count of even numbers.

The first even number greater than or equal to 1331 is 1332.
The last even number less than or equal to 3113 is 3112.

Count of even numbers in the range = (3112 - 1332)/2 + 1 = 893

Count of Numbers Divisible by 3:
To find the count of numbers divisible by 3 in the given range, we need to find the count of numbers that leave a remainder of 0 when divided by 3.

The first number greater than or equal to 1331 that leaves a remainder of 0 when divided by 3 is 1332.
The last number less than or equal to 3113 that leaves a remainder of 0 when divided by 3 is 3111.

Count of numbers divisible by 3 in the range = (3111 - 1332)/3 + 1 = 594

Count of Numbers Divisible by 5:
To find the count of numbers divisible by 5 in the given range, we need to find the count of numbers that leave a remainder of 0 when divided by 5.

The first number greater than or equal to 1331 that leaves a remainder of 0 when divided by 5 is 1335.
The last number less than or equal to 3113 that leaves a remainder of 0 when divided by 5 is 3110.

Count of numbers divisible by 5 in the range = (3110 - 1335)/5 + 1 = 555

Count of Numbers Divisible by 2, 3, or 5:
To find the count of numbers divisible by 2, 3, or 5 in the given range, we need to find the count of numbers that are divisible by at least one of these numbers.

We can use the principle of inclusion-exclusion to find this count.

Count of numbers divisible by 2, 3, or 5 = Count of numbers divisible by 2 + Count of numbers divisible

Answer

Ans is 477.

For total nos between 1331 and 3113 = 3113-1331 -1 = 1781 (excluding end nos)

Of 1781 nos , the nos divisible by 2 = 1781 +1 = 891 (as the starting no is even so no of even nos is greater than no of odd nos )

now nos which aren't divisible by 2 = 890

of 890 nos, nos which are divisible by 5 but don't end in 0 = = 1781/10= 178

(as the nos ending in 0 are already eliminated)

(for every 10 nos there is one no divisible by 5 and not ending in 0)

so now, nos not divisible 2 or 5 = 890 -178 = 712

now for nos divisible by 3, take the nos from 1338 to 3108 which has 1770 nos

for every 30 nos , there are 10 nos divisible by 3. of which 5 are also divisible by 2 and one is divisible by 5 but not ends in 0. So for every 30 nos , 4 nos are divisible by 3 but not by 2 or 5

so of 1770 nos there are 59 sets of 30 nos . hence 4 * 59 = 236 nos are divisible by 3 but not by 2 or 5

so nos not divisible by 2 or 3 or 5 in given range = 712 - 236 = 476

but we haven't taken 3 multiples in range 1331 to 1338 and 3108 to 3113

nos multiple of 3 in this range is = 1334 (even and hence eliminated) , 1335 (divisible by 5 and ends in 5) , 3111(not eliminated)

so 476 - 1 = 475

but i have excluded end nos so adding 2 = 475 + 2 = 477

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