12 Days of Christmas GMAT Competition - Day 6: If x = 2^16*3*3*5^18 +

Question

12 Days of Christmas GMAT Competition with Lots of Fun

If  x = 2^{16}*3^3*5^{18} + 9^{15} , then how many digits does x have?

A. 16
B. 17
C. 18
D. 19
E. 20

https://gmatclub.com/blog/wp-content/uploads/2020/12/12-days-of-christmas.jpg

M37-41

vivek1408 · Accepted Answer

X can be rewritten as 10^16*225 + 9^15
Or x= 2.25*10^18 + 9^15

The left hand term has 19 digits, right hand term is lesser than 15 terms, so adding it to a 19 digit number won’t change the number of digits of the total.

Hence answer is (D)19

Posted from my mobile device

Bunuel · Answer

Official Solution:

If  x = 2^{16}*3^3*5^{18} + 9^{15} , then how many digits does  x  have?

A.  16 
B.  17 
C.  18 
D.  19 
E.  20

First, let's determine the number of digits in the first term:

2^{16}*3^3*5^{18}= 
 
 =(2^{16}*5^{16})*3^3*5^2= 
 
 =10^{16}*675

This number will have 19 digits, consisting of 675 followed by 16 zeros.
 
Next, we need to determine if adding  9^{15}  will increase the number of digits. Notice that  9^{15} < 10^{15} . If we replace the second term with  10^{15} , which is greater than  9^{15} , we would have:

2^{16}*3^3*5^{18}+10^{15}= 
 
 =10^{16}*675+10^{15}= 
 
 =10^{15}(10*675+1)= 
 
 =10^{15}*6751 .

This number will also have 19 digits, specifically 6751 followed by 15 zeros.
 
Since adding  10^{15} , which is greater than  9^{15} , does not increase the number of digits, it implies that adding  9^{15}  will not increase the number of digits either. Therefore, the number  x = 2^{16}*3^3*5^{18} + 9^{15}  will also have 19 digits.

Answer: D

dep · Answer

x = 2 ^{16}*3^3*5^{18}+9^{15}

x = 10 ^{18}(\frac{27}{4})+(10-1)^{15}

x= 10 ^{18}(6.9+\frac{10^{15}(1-.1)}{10^{18}})

x = 6.9 * 10^{18}

Total Digits = 19

IMO D

DJ2911 · Answer

Answer: D(19)

2^16*3^3*5^18 + 9^15
6750 * 10^15 + 9^15
Since 10^15 > 9^15
6750 * 10^15 >>>9^15
Clearly the value on the right is at least 6750 times smaller and will thus have no consequence on the number of digits in the overall sum.
Digits in 6750*10^15 = 19
Thus answer is 19

Posted from my mobile device

Eswar42 · Answer

D is the correct answer  
there are two arithmetic expressions are added,
1. 2^16 * 3^3 * 5^18 = (2*5)^16 * 3^3 * 5^2 = 675 * 10^16 = 19 digits
2. 9^15 = (10-1)^15, is a number which is having 15 digits
1+2 will give out a number with 19 digits for sure. So correct answer is D

hunter2000 · Answer

first join 2 and 5 to form 10 
so now we have 
10^16 * 25 * 27 + 9 ^15

10^16 * 675 + 9 ^15

note that 10^ 16 is lot greater than 9^15

so the answer is 67500..... ( 19 digits ) + 9^15

so total 19 digits .

answer is D

MBAB123 · Answer

X = 2^16 * 3^3 *5^18 + 9^15. 
-> Lets look at the first part before the "+". It can be written as x = (2^16*5*16)*(3^3*5^2)
=> (10^16) (27*25)
=> 675*10^16. (Now this number will have 19 digits and the first 3 digits will be 675 followed by 16 zeroes.

Now lets look at 9^15. For our own sake, think of 10^15.
10^15 is a 16 digit number starting with 1 and followed by 15 zeros. 
675*10^16 + 10^16 = Don't have to solve this. It is evident that the number of digits will not change.

If 10^15 is not changing the number of digits, obviously, 9^15 will not change it.

Hence, the answer is D-19.

vv65 · Answer

My answer is D (19)

can be simplified 
to  , which has 19 digits

is less than  , and so it will have 15 digits

The sum of the two will have 19 digits

Posted from my mobile device

Aks111 · Answer

x =  2^16 * 3^3 * 5^18 + 9^15 
=  2^16 * 5^16 * 5^2*27 + 9^15 
=  10^16*25*27 + 9^15 
=  10^16*25*(25+2) + 9^15 
=  10^16*(625+50) + 9^15 
=  10^16*675 + 9^15

Now,  9^15  will be smaller than  10^15  (1 followed by 15 zeros) and thus less than 16 digits.
 10^16*675  is 675 followed by 16 zeros. Total 19 digits.
Hence, answer  D.

Poojita · Answer

9^15 = (10-1)^15, is a number which is having 15 digits 
how did you arrive at this part?

pudu · Answer

from (10-1)^15 how do you get 10^15(1-1)?please explain i'm unable to understand it...

bumpbot · Answer

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.

12 Days of Christmas GMAT Competition - Day 6: If x = 2^1633*5^18 +

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

12 Days of Christmas GMAT Competition - Day 6: If x = 2^16*3*3*5^18 +

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

12 Days of Christmas GMAT Competition - Day 6: If x = 2^1633*5^18 +