GMAT Club World Cup 2022 (DAY 10): Is the sum of four consecutive

Question

Is the sum of four consecutive integers a multiple of 10 ?

(1) The product of the four integers is a multiple of 120.
(2) The product of the units digits of the four integers is a multiple of 120.

Iotaa · Answer

Assume the integers be I, I+1, I+2, I+3. Question is 4I+7 is a multiple of 10 or not

(1) The product of the four integers is a multiple of 120: 120 is 2*2*2*3*5, or 2*3*4*5. So for any value, summation will not be a multiple of 10.  Sufficient

(2) The product of the units digits of the four integers is a multiple of 120: numbers can be 12,13,14,15; 22,23,24,25; 32,33,34,35 ..... Again for any value, summation will not be a multiple of 10.  Sufficient

D is the correct option

Kinshook · Answer

Asked: Is the sum of four consecutive integers a multiple of 10 ?

Let the consecutive integers be x, x+1, x+2 & x+3.
Is (x+x+1+x+2+x+3) = 4x + 6 is a multiple of 10 ?

(1) The product of the four integers is a multiple of 120.
x(x+1)(x+2)(x+3) = (x^2 + 3x)(x^2 + 3x+2) = (t-1)(t+1) = t^2 - 1; where t = x^2 + 3x +1
t^2 - 1 = 120k; t^2 - 120k - 1 = 0;  t = \sqrt{120k+1}  or  t = -\sqrt{120k+1} 
If k =1; t^2 = 121; For other values of k, t is NOT an integer. Therefore, x is NOT an integer.
t=x^2 + 3x +1 = 11; x^2 + 3x -10 = 0; (x+5)(x-2) = 0; x = 2 or x = -5
t=x^2 + 3x +1 = -11; x^2 + 3x +12 = 0; x is NOT an integer
Consecutive integers = {-5,-4,-3,-2} or {2,3,4,5}
Sum of 4 integers = (2+3+4+5) = 14 or -(2+3+4+5) = -14; 
Sum of 4 integers is NOT a multiple of 10.
SUFFICIENT

(2) The product of the units digits of the four integers is a multiple of 120.
2*3*4*5 = 120 = 120*1 : Sum of unit digits = 2+3+4+5 = 15: NOT a multiple of 10
3*4*5*6 = 360 =120*3: Sum of unit digits = 3+4+5+6 = 18: NOT a multiple of 10
4*5*6*7 = 840 = 120*7: Sum of unit digits =4+5+6+7 = 22: NOT a multiple of 10
5*6*7*8 = 1680 = 120*14: Sum of unit digits = 5+6+7+8 = 26: NOT a multiple of 10
6*7*8*9 is NOT a multiple of 120
7*8*9*0 = 0 = 120*0: Sum of unit digits =7+8+9+0 = 24: NOT a multiple of 10
SUFFICIENT

IMO D

gmatophobia · Answer

Statement 1

120 = 5 * 3 * 2^3

This indicates that out of the four consecutive numbers, one number should be a multiple of 5.

The number 5 can come in any position, i.e. the series can either start with 5 or end with 5 or 5 can occupy any of the two middle positions.

Ex.

5 6 7 8
4 5 6 7 
3 4 5 6
2 3 4 5

Alternatively, we can also have 10

10 11 12 13
09 10 11 12
08 09 10 11
07 08 09 10

To ascertain "Is the sum of four consecutive integers a multiple of 10 ?", we only care about the last digit. Hence in both the sets we see that the sum of the last digits is not a multiple of 10.

Hence this statement is enough to ascertain that the sum is not a multiple of 120

Statement 2

The analysis to this is similar to what we have already done. Infact, its a subset -

The number can have the unit digit in any one of the below series -

5 6 7 8
4 5 6 7 
3 4 5 6
2 3 4 5

As discussed in statement 1 , the sum of the unit digits does not add to be a multiple of 10, hence this statement is also sufficient

IMO D

sanjitscorps18 · Answer

Is the sum of four consecutive integers a multiple of 10 ?

(1) The product of the four integers is a multiple of 120.
If we go through the product of four numbers we get a pattern and this pattern continues
2,3,4,5 -->  Product multiple of 120  ---  Sum not multiple of 10 
3,4,5,6 -->  Product multiple of 120  ---  Sum not multiple of 10 
4,5,6,7 -->  Product multiple of 120  ---  Sum not multiple of 10 
5,6,7,8 -->  Product multiple of 120  ---  Sum not multiple of 10 
6,7,8,9 -->  Product is NOT multiple of 120  ---  Sum IS multiple of 10 
7,8,9,10-->  Product multiple of 120  ---  Sum not multiple of 10 
8,9,10,11-->  Product multiple of 120  ---  Sum not multiple of 10 
9,10,11,12 -->  Product multiple of 120  ---  Sum not multiple of 10 
10,11,12,13 -->  Product multiple of 120  ---  Sum not multiple of 10 
11,12,13,14 -->  Product is NOT multiple of 120  ---  Sum IS multiple of 10 
12,13,14,15 -->  Product multiple of 120  ---  Sum not multiple of 10 
13,14,15,16 -->  Product multiple of 120  ---  Sum not multiple of 10 
14,15,16,17 -->  Product multiple of 120  ---  Sum not multiple of 10 
15,16,17,18 -->  Product multiple of 120  ---  Sum not multiple of 10 
16,17,18,19 -->  Product is NOT multiple of 120  ---  Sum IS multiple of 10

Answer is NO. Sufficient

(2) The product of the unit digits of the four integers is a multiple of 120.
Taking the same pattern from above we get
0,1,2,3 -->  Product multiple of 120  ---  Sum not multiple of 10 
2,3,4,5 -->  Product multiple of 120  ---  Sum not multiple of 10 
3,4,5,6 -->  Product multiple of 120  ---  Sum not multiple of 10 
4,5,6,7 -->  Product multiple of 120  ---  Sum not multiple of 10 
5,6,7,8 -->  Product multiple of 120  ---  Sum not multiple of 10 
7,8,9,0-->  Product multiple of 120  ---  Sum not multiple of 10 
8,9,0,1-->  Product multiple of 120  ---  Sum not multiple of 10 
9,0,1,2 -->  Product multiple of 120  ---  Sum not multiple of 10

Answer is NO. Sufficient

IMHO Option D

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

GMAT Club World Cup 2022 (DAY 10): Is the sum of four consecutive

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 Data Sufficiency (DS) Questions

GMAT Club World Cup 2022 (DAY 10): Is the sum of four consecutive

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