The first 23 natural numbers are written in increasing order

Question

The first 23 natural numbers are written in increasing order beside each other to form a single number. What is the remainder when this number is divided by 18? 
 
(A) 1 
 
(B) 6 
 
(C) 12 
 
(D) 15 
 
(E) 18

Posted from my mobile device

Kinshook · Answer

Given:  The first 23 natural numbers are written increasing order beside each other to form a single number.

Asked:  What is the remainder when this number is divided by 18?

18 = 2 * 3^2 = 2*9

The number N = 1234567891011121314151617181920212223

Divisibility by 9:
Sum of digits = 1+2+3+ .....23 = 23*24/2 = 23*12 = 276
Remainder when divided by 9 = 276mod9 = 6
Nmod9 = 6
N = 9k +6

Divisibility by 2:
The number = 1234567891011121314151617181920212223
When divided by 2, leaves remainder = 1
Nmod2 =1
N = 2m +1
N = 9k + 6 = 2m + 1
Since 2m+1 is odd => k = odd = 2x+1
N = 9(2x+1) + 6 = 18x +9+6 = 18x +15

IMO D

Kinshook · Answer

When N = 6mod9 & N=1mod2
Is there is better way to find N = xmod18 ?

GMATinsight · Answer

First Comment: GMAT does NOT use the terms such as  natural numbers  which clearly indicates that it's not from a GMAT practice source or not from a reputed or informed GMAT practice source

for GMAT, Natural number =  Positive Integers

The solution is as attached

lacktutor · Answer

I think the best way to solve this question is to check the answer choices.

—> 1234...212223 is ending with odd number(...3)—> when you divide this odd number by 18, the remainder will be odd too.

—> that means that we need to eliminate the answers B,C and E.

Let’s check the answer A:
—> In order the number to be divisible by 18, the number should be divided by 2 and 9 at the same time.

—> If the remainder is 1, the last digit of the number will be 2(1234...20212222)
—> new number is divisible by 2, but not by 9. 
Because 
—> When you sum up all the digits of the number, the amount will be 113–> 113 is not divided by 9
(According to the Division rules for 9, if the sum of the digits is divisible by 9, then the actual number is also divisible by 9.)
A is out.

We have only an answer choice D.

(If the remainder is 15, the number will be 1234....20212208–> this new number is divisible by 2. 
—> the sum of the digits of new number is 117. It is divisible by 9.)

The answer is D

Posted from my mobile device

gracie · Answer

cycle of remainders when 18 divides into integer with units digit of 3, starting with 23:
5,15,7,17,9,1,11,3,13,5,15
15 is only possible option
D

AcetheGMAT2019 · Answer

Since 2m+1 is odd => k = odd = 2x+1
N = 9(2x+1) + 6 = 18x +9+6 = 18x +15

Did not understand this bit. Can someone pls help ?

Hovkial · Answer

The term "natural numbers" is widely used in mathematics scholarship, research, journals, etc. It is the basis on which the term "integer" was developed. 
 
Some mathematicians use the term natural numbers to include the number zero and some start with the number one. Due to this disagreement, mathematics have tended to order using the symbol N followed by small zero to denote the set of natural numbers that include zero and the symbol N followed by small 1 to denote the alternative case.
 
The "set" of natural numbers is originally defined in historical literature where names of eminent scientists, thinkers, writers teachers and philosophers feature. Without the extensive work on natural numbers, modern mathematics would not have developed the concept of the "integer" and subsequent terminology -- positive and negative integers. 
 
One of the purposes of this forum is to educate users. But now we have a self-appointed "GMAT expert" telling us that the question is irreputable and uninformed because it used the term natural number. The term was included to remain faithful to the original. 
 
I do not expect a whole lot from prep companies and self-appointed, profit-oriented "GMAT expert" who daily peddle their wares as the one and best solution to hapless GMAT testtakers. But it would certainly be nice if they refrained from insulting history, academics, writers, thinkers, philosophers and the many others of us who strive to hold high scholarship standards.

Posted from my mobile device

Hovkial · Answer

He had written: 
 
N = 9k+6 = something odd. 
 
The only way you can make the quantity on the left-hand side odd would be to make the term "k" odd. 
 
This is because 6 is even, implying you need an odd term in 9k, which in turn is an odd number 9 multiplied by an integer k.

Posted from my mobile device

AcetheGMAT2019 · Answer

Hovkial

Thanks

Sent from my GM1901 using Tapatalk

Bunuel · Answer

GMATinsight is right. You are not.

No GMAT official question used "natural number". Instead official questions use positive integer.

One reason for that is the term natural number in some definitions include 0 and in other don't.

The first 23 natural numbers are written in increasing order

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

The first 23 natural numbers are written in increasing order

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