Multiples of 11 between 100 and 500 ---- 37 or 36? :-(

Shaikha wrote:Number of integers = (Biggest - Smallest)/(distance between each successive pair) + 1

This formula always holds only when the "Biggest" and "Smallest" are both multiple of the "distance" in question. If they are not, then sometimes it will hold and sometimes it will not.

Let me clarify with some example,
Number of multiples of 11 between 1 and 12 is 1 but using the formula it is (12 - 1)/11 + 1 = 2
Number of multiples of 11 between 2 and 12 is 1 and using the formula it is (12 - 2)/11 + 1 = 1

In the first case, the formula yields wrong answer but in the second case it yields correct answer.

Hence, it is always recommended to find out the smallest and largest multiple first, then use the formula.

In our case,

• Smallest multiple of 11 between 100 and 500 = 110
  Largest multiple of 11 between 100 and 500 = 495
  
  Hence, number of multiples of 11 between 100 and 500 = (495 - 110)/11 + 1 = 385/11 + 1 = 35 + 1 = 36

Shaikha wrote:The number of integers between 100 and 500 that are multiples of 11 is required.

Here's another approach.

You want to determine the number of terms in the sequence {110, 121, 132, . . . , 484, 495}

Observe that:
110 = (10)(11)
121 = (11)(11)
132 = (12)(11)
.
.
.
484 = (44)(11)
495 = (45)(11)

So, the question now becomes, "how many integers are there from 10 to 45 inclusive?"

The rule says: the number of integers from x to y inclusive equals y - x + 1

So, the number of integers from 10 to 45 inclusive = 45 - 10 + 1 = 36

Cheers,
Brent