Five integers between 10 and 99, inclusive, are to be formed by using

Question

Five integers between 10 and 99, inclusive, are to be formed by using each of the ten digits exactly once in such a way that the sum of the five integers is as small as possible. What is the greatest possible integer that could be among these five numbers?

A. 98
B. 91
C. 59
D. 50
E. 37

PS30402.01

A. 98
B. 91
C. 59
D. 50
E. 37

PS30402.01

Shobhit7 · Accepted Answer

To minimize the sum of 5 two-digit integers created by using digits 0-9 without repetition,

largest digits must be in unit place: 9,8,7 and 6
integer 0 can only be placed in units place in order to create a 2 digit integer.

so, integers left for tens place are: 1,2,3,4 and 5

Sum of integers in units place is: 9+8+7+6+0= 30
Sum of integers created with tens digit is: 50+40+30+20+10= 150

So, sum of five integers will always be 30+150 = 180 irrespective of wherever we place these single digit integers.

In such a case, largest possible 2 digit integer is 59.

Hence, Ans is choice C

BrentGMATPrepNow · Answer

To minimize the sum of the five integers, we will make the tens digits as smallest possible. 
This means we'll use 1, 2, 3, 4, and 5 as the TENS digits.

So, for the moment, the five integers look like this: 1_, 2_, 3_, 4_, and 5_

At this point we need to use the other five possible digits (0, 6, 7, 8, and 9) for the UNITS digits

IMPORTANT: It doesn't matter where we place these units digits. 
For example, if we make our five integers 10, 26, 37, 48, and 59, their sum is 180
Alternatively, if we make our five integers 19, 20, 38, 46, and 57, their sum is still 180
And, if we make our five integers 17, 28, 39, 46, and 50, their sum is still 180
etc..

The question asks "What is the  greatest  possible integer that could be among these five numbers?"
So, the greatest possible integer value is 59

Answer: C

Cheers,
Brent

Cebe3004 · Answer

The Tens digits have to hold values   & Units digits have to hold  
It can be {10, 29, 38, 47, 56}, {19, 28, 37, 46, 50} or {10, 26 37, 48, 59}. In all the cases the sum is 180, which is the least possible sum. 
However, the 3rd set has the greatest integer which is  59 -> Answer

ScottTargetTestPrep · Answer

SInce we want the sum of the 5 two-digit numbers to be as small as possible, we want the tens digits of each number to be as small as possible. Since the tens digit can’t be 0, the five smallest non-zero digits are 1, 2, 3, 4, 5. Therefore, the largest possible integer is in the 50s. How we pair 0, 6, 7, 8 and 9 (as the units digits) with 1, 2, 3, 4, 5 (as the tens digits) doesn’t matter. For example, the sum of 10, 26, 37, 48 and 59 is equal to the sum of 19, 28, 37, 46 and 50 (notice that either sum is 180). Therefore, the greatest possible integer is 59.

Answer: C

GMATGuruNY · Answer

To MINIMIZE the sum of the five numbers, we must minimize the TENS digits for the five numbers.
How?
By using the smallest possible options ( 1, 2, 3, 4, 5 ) for the tens digits and the remaining options ( 0, 6, 7, 8, 9 ) for the units digits.
No matter how the 5 blue tens digits are combined with the 5 red units digits to form the five numbers, the following sum will be yielded:
( 10+20+30+40+50 ) + ( 0+6+7+8+9 ) = 180

What is the greatest possible integer that could be among these five numbers? 
If we combine the largest blue tens digit ( 5 ) with the largest red units digit ( 9 ), we get:
 5  9

C

Archit3110 · Answer

So as to have the the sum of the five integers is as small as possible.
The tens digit will have to be ( 1,2,3,4,5) & Units ( 6,7,8,9,0)
max value will be 59
IMO C

HasnainAfxal · Answer

To have the sum as small as possible and to choose numbers with different tenth digit, we can choose number 10, 20, 30, 40 and 50 so the smallest possible sum is 150 and greatest possible number is 50.

Let me know please where i am doing wrong?

Let me know please where i am doing wrong?

sunny91 · Answer

Shobhit7 , repetition is not mentioned in the units place. so, how can we assume that.

phuulinh225 · Answer

I can't find where in the question states that the unit digits must be distinct. Could you help clarify? Many thanks.

carma19 · Answer

"Five integers between 10 and 99, inclusive, are to be formed by using each of the   ten   digits  exactly once "
The ten digits are: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. 
So, in order to have " the sum of the five integers as small as possible " we have to use:

a) 1, 2, 3, 4, 5 for the tenth-position: 10+20+30+40+50=150
b) 9, 8, 7, 6, 0 (the remaining digits out of the ten mentioned above) for the unit-place: 9+8+7+6+0=30

TOT = 180
Largest two-digit integer:  59 .

Hence, (C)

aarushisingla · Answer

To have the sum as small as possible and to choose numbers with different tenth digit, we can choose number 10, 20, 30, 40 and 50 so the smallest possible sum is 150 and greatest possible number is 50.

Let me know please where i am doing wrong?

Let me know please where i am doing wrong as it is nowhere mentioned that units digit must be different.

alexkors · Answer

any digit must be used exactly once

KarishmaB · Answer

You need to use all the digits from 0 - 9 (only once) but the integers must be 2 digit integers. So 0 cannot be in tens place.
The tens place must be as small as possible to minimise the sum so we should use 1, 2, 3, 4 and 5 in tens place. 
The units places of these 5 numbers will be taken by 6, 7, 8, 9 and 0. Note that it doesn't matter where we place which units digit, the sum will stay the same:

10 + 26 + 37 + 48 + 59 = Sum

16 + 27 + 38 + 49 + 50 = Same Sum

19 + 28 + 37 + 46 + 50 = Same Sum

because in each case, we are just adding 10 + 20 + 30 + 40 + 50 + 6 + 7 + 8 + 9 + 0.

This sum will be the lowest. The maximum value of one of the five numbers then will be 59.

Rukia · Answer

I was confused as well initially but I realized after reading the question again few times that it wants you to use the ten digits(0,1,2,3,4,5,6,7,8,9) not the TENTH digit only ONCE to form the two digit integers. Hope this helps.

faat99 · Answer

As a disclaimer, I got this question wrong, but I do not agree with all the answers her, and a bit disappointed that all the GMAT experts trying to justify that C is the correct answer.

Unlike all the people's attempted explanation here that "we can't use the digit more than once", there is no where in the question assumes that the digits have to be used only once.

The only condition is:  five integers, formed by using the ten digits exactly once, such that the sum is as small as possible

Base on this condition, these five integers are: 10;20;30;40;50

You got to think critically, rather than blindly reciting the OG answer. Happy to know if I've missed anything.

Bunuel · Answer

Please read the question carefully: "Five integers between 10 and 99, inclusive, are to be formed by   using each of the ten digits exactly once   in such a way that the sum of the five integers is as small as possible. What is the greatest possible integer that could be among these five numbers?"

faat99 · Answer

Even in consideration of the highlight - that still doesn't change my statement:
10,20,30,40,50. the tenth digits of these five integers are 1,2,3,4,5, used only once.

Bunuel · Answer

Again you are misunderstanding the question completely.

I'll try again for the last time. There are ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The question asks to form FIVE two-digit numbers  using each of the ten digits exactly once  in such a way that the sum of the five integers is as small as possible.

In your example, digit 0 is used more than once. We are explicitly told that we should use each of the ten digits exactly once!

If you use 1, 2, 3, 4, and 5 as the tens digits of five numbers: 1_, 2_, 3_, 4_, and 5_, then for the five units digit you must use the remaining digits 0, 6, 7, 8, and 9 ONCE. So, the greatest possible integer value is 59.

P.S. FYI, among thousands of official quant questions (Paper test, GMAT Prep, OG's...) I can recall one or two questions which were wrong. ALL other official questions are correct.

faat99 · Answer

Hey Bunuel, my apologies you are totally right i have misread the question, it is ten digit once not tenth digit. sorry for the confusion

Five integers between 10 and 99, inclusive, are to be formed by using

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Five integers between 10 and 99, inclusive, are to be formed by using

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