Question

How many positive 3-digit numbers are multiples of 20, but not of 55?

Answer 1

Hint: According to the question given in the question we have to determine the positive 3-digit numbers are multiples of 20, but not of 55. So, first of all to determine the positive 3-digit numbers are multiples of 20, but not of 55 therefore numbers to be a multiple of 20, a natural number would have to be multiplied by 20 to get it and for instance, 40 is a multiple of 20.
Now, we have to check for the numbers which are multiple of 20 are 3-digit numbers. The lowest 3-digit number that should be multiple of 20.
Now, for a number to be both a multiple of 20 and 55 we have to determine the L.C.M by prime factorization.
Now, we have to determine the every multiple after that number will be a required multiple of the number 220.
Now, after obtaining the numbers that are multiples of both 20 and 55 we have to determine that how are multiples of 20 and not 55, so we have to subtract the result obtained from our multiples of 20 results.

Complete step-by-step solution:
Step 1: First of all to determine the positive 3-digit numbers are multiples of 20, but not of 55 therefore numbers to be a multiple of 20, a natural number would have to be multiplied by 20 to get it and for instance, 40 is a multiple of 20. Hence,
\[ \Rightarrow 20 \times 2 = 40\] and it is not 66.
$
   \Rightarrow 20 \times 3 = 60 \\
   \Rightarrow 20 \times 4 = 80
 $
So, the same goes for 55 and its multiplies.
Step 2: Now, we have to check for the numbers which are multiple of 20 are 3-digit numbers. The lowest 3-digit number that should be multiple of 20. Hence,
$ \Rightarrow 20 \times 5 = 100$
And the largest is 980.
$ \Rightarrow 20 \times 49 = 980$
Which is $49 - 5 + 1 = 45$ multiplies.(In which we add 1 to include the first multiple).
Step 3: Now, for a number to be both a multiple of 20 and 55 we have to determine the L.C.M by prime factorization. Hence,
$
   \Rightarrow 20 = 2 \times 2 \times 5 \\
   \Rightarrow 55 = 5 \times 11
 $
So, the L.C.M is,
$ \Rightarrow 2 \times 2 \times 5 \times 11 = 220$
Now, we have to determine every multiple after that will be a multiple of 220,
\[
   \Rightarrow 220 \times 2 = 440 \\
   \Rightarrow 220 \times 3 = 660 \\
   \Rightarrow 220 \times 4 = 880
 \]
Step 4: Now, after obtaining the numbers that are multiples of both 20 and 55 we have to determine that how are multiples of 20 and not 55, so we have to subtract the result obtained from our multiples of 20 results.
$ \Rightarrow 45 - 4 = 41$

Hence, we have determined positive 3-digit numbers are multiples of 20, but not of 55 which are 41.

Note: After the multiple of 880 the next multiple will be a four digit number and so we do not want to determine the four digit number as it is not asked in the question.
To determine the required numbers it is necessary that we have to determine the L.C.M or we can say that the factors of the numbers which are 20 and 55.