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

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

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