# Number Properties Puzzle – Challenge 11

This is another great math challenge for those who love critical thinking challenges. This puzzle doesn’t require advanced mathematical knowledge, just logical reasoning. Let’s see how smart you are!

## Challenge:

How many positive integers less than 1000 do not have any 5 as any digits?

A- 700

B- 728

C- 736

D- 770

E- 819

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The correct answer is B.

Let’s consider 1 digit, 2 digits, and 3 digits numbers separately:
One digit number: From 9 1-digit numbers, eight numbers don’t have any digit as 5.
Two digits numbers: Put A for $$10^{th}$$ place and B for unit place. Then, we have AB as a 2 digits number. We can put 8 digits for A (all digits except 0 and 5) and 9 digits for B (except 5). So, we’ll have 8 × 9 = 72 2-digits numbers that don’t have 5 as any digits.
For three digits numbers, ABC: For A, we can put 8 digits (not 0 or 5), for B, we have 9 digits (not 5) and for C, we have 9 digits (not 5). So, there are 648 three digits numbers that don’t have 5 as any digits. 8 × 9 × 9 = 648
In total: 8 + 72 + 648 = 728

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