Max@Math Revolution wrote:
How many 5 digit numbers have at least one zero digit?
A. 30951
B. 40141
C. 47132
D. 50001
E. 50433
NOTE: We can solve this question without performing lengthy calculations (e.g., 9^5)
# of 5-digit numbers with at least one zero digit = (TOTAL number of 5-digit numbers) - (number of 5-digit numbers without ANY 0's)
As Keith shows us above, we get:
# of 5-digit numbers with at least one zero digit = 90,000 - 9^5
Before we calculate 9^5 (and then subtract that value from 90,000), we should keep in mind that the GMAT test-makers are NOT interested in our ability to perform lengthy/tedious calculations (for more on this, read
https://www.gmatprepnow.com/articles/re ... st-maker-0)
We already know that the answer to the question is 90,000 - 9^5
Notice that this value is a multiple of
9, since 90,000 - 9^5 =
9(10,000 - 9^4)
So, let's check the answer choices so see which one (or ones) is divisible by
9
RULE: If a number is divisible by 9, then the sum of its integers is divisible by 9.
A. 30951 Sum of digits = 18, which is divisible by 9. Since, answer choice A is divisible by 9, we'll KEEP IT.
B. 40141 Sum of digits = 10, which is NOT divisible by 9. Since, answer choice B is NOT divisible by 9, we'll ELIMINATE IT.
C. 47132 Sum of digits = 17, which is NOT divisible by 9. Since, answer choice C is NOT divisible by 9, we'll ELIMINATE IT.
D. 50001 Sum of digits = 6, which is NOT divisible by 9. Since, answer choice D is NOT divisible by 9, we'll ELIMINATE IT.
E. 50433 Sum of digits = 15, which is NOT divisible by 9. Since, answer choice E is NOT divisible by 9, we'll ELIMINATE IT.
By the process of elimination, the correct answer is A
Cheers,
Brent
