gmattesttaker2 wrote:Hello,

Can you please help with this:

What is the sum of the digits of the positive integer n where n < 99?

1) n is divisible by the square of the prime number y.
2) y4 is a two-digit odd integer.

Answer given is C

1) If, n = 25 => 25/(5.5)
n = 49 => 49/(7.7)

Insufficient. However, I am not able to proceed after this. Can you please assist? Thanks a lot.

Best Regards,
Sri

1: if prime number is 2, 2^2=4.
If n=16, n is divisible by 4, sum of digits =7
If n=20, n is divisible by 4, sum = 2
Hence Insufficient.

2: y^4 is a two-digit odd integer.
Doesn't tell us anything about n. Insufficient.

Together:
Smallest odd integer is 3. 3^4 = 81. Two digit and odd.
Next one is 5. 5^4 = 625. This is a 3 digit integer.
Therefore y=3.
Now n is divisible by y^2=3^2 = 9.
Now the cool thing is that if a 2 digit number, less than 99, is divisible by 9, the sum of its digits is always 9. (9, 18, 27......81 etc. all add upto 9).
Hence sum of digits of n is 9. Sufficient.

Hence C.

A great example of a problem where knowing number properties is a great shortcut!

eagleeye wrote: Now the cool thing is that if a 2 digit number, less than 99, is divisible by 9, the sum of its digits is always 9. (9, 18, 27......81 etc. all add upto 9).

Also, this is true for all multiples of 9 regardless of number of digits. 279 = 2 + 7 + 9 = 18.

eagleeye wrote:

gmattesttaker2 wrote:Hello,

Can you please help with this:

What is the sum of the digits of the positive integer n where n < 99?

1) n is divisible by the square of the prime number y.
2) y4 is a two-digit odd integer.

Answer given is C

1) If, n = 25 => 25/(5.5)
n = 49 => 49/(7.7)

Insufficient. However, I am not able to proceed after this. Can you please assist? Thanks a lot.

Best Regards,
Sri

1: if prime number is 2, 2^2=4.
If n=16, n is divisible by 4, sum of digits =7
If n=20, n is divisible by 4, sum = 2
Hence Insufficient.

2: y^4 is a two-digit odd integer.
Doesn't tell us anything about n. Insufficient.

Together:
Smallest odd integer is 3. 3^4 = 81. Two digit and odd.
Next one is 5. 5^4 = 625. This is a 3 digit integer.
Therefore y=3.
Now n is divisible by y^2=3^2 = 9.
Now the cool thing is that if a 2 digit number, less than 99, is divisible by 9, the sum of its digits is always 9. (9, 18, 27......81 etc. all add upto 9).
Hence sum of digits of n is 9. Sufficient.

Hence C.

Hello Eagleeye,

Thank you very much for your explanation. As always, it is very detailed and clear. Thanks a ton.
@Bill, thanks for your tip as well.

Best Regards,
Sri