If the tens digit of the positive integer x and the tens digit of the

Question

If the tens digit of the positive integer x and the tens digit of the positive integer y are both 6, what is the number of different possible values for the tens digit of 2x + 2y?

A. Three
B. Four
C. Five
D. Six
E. Seven

Show Spoiler

Attachment:

Screenshot 2024-01-02 202125.png

chetan2u · Accepted Answer

The tens digit of 2x+2y will depend only on the units digit, so it doesn’t matter what other digits are. 
Let the two numbers be 6a and 6b. 
Least value of 2(x+y) = 2(60+60) = 240
Maximum value of 2(x+y) = 2(69+69) = 276

So 2(x+y) can take all even values from 240 to 276. 
Thus, tens digit can be 4, 5, 6 and 7, a total of four values.

B

gmatophobia · Answer

x = 6p  ⇒  60 + p

y = 6q  ⇒  60 + q

2x = 120 + 2p

2y = 120 + 2q

2x + 2y = 240 + 2(p+q)

The minimum value of  p + q = 0 , and the maximum value of  p + q = 18

Hence, the minimum value of  2(p+q) = 0 , the maximum value of  2(p+q) = 36 . From this information, we can infer that while performing the addition operation we can have a carryover of 0, 1, 2, or 3 from the unit place to the tens place.

Possible values of the tens place = (4+0), (4+1), (4+2), (4+3) = 4, 5, 6, or 7.

Option B

tickledpink001 · Answer

and the maximum value of p+q=18 
why's this the case please?

gmatophobia · Answer

tickledpink001  -  p  and  q  are single digits non-negative integers (i.e. both  p  and  q  can take values from  0  to  9 , inclusive). Hence, the maximum value that  p  and  q  can have is  9 , resulting in the maximum value of  p + q = 18 .

Hope this helps.

versewriter · Answer

We have to find the unique tens digit number from 2x + 2y

Min number could be 61 and max number could be 69.

So the range of numbers from 240 to 276 which we have to find the unique number of tens digits.

Only unique digits we will get from range 240 to 276 is 4 , 5, 6, 7 as unique tens digits

So total 4 unique digits.

Option - B will be the answer

Engineer1 · Answer

Why did you assume the integer is a 2 digit integer? Thank you.

DmitryFarberMPrep · Answer

We don't know the number of digits, but it doesn't matter. When we add, no digit affects the value of those to its right, only those to its left. 
For instance, if we add 2169 + 53,869, the answer still ends in 38, just as when we add 69 + 69. Similarly, when we double the value, those additional places to the left don't change the fact that 38 doubles to 76.

So in short, we can focus only on the relevant digits and not worry about any other potential digits. This is common on place value/digit problems.

Engineer1 · Answer

Understood, thank you. I think you meant this, correct?  " When we add, no digit affects the value of those to its  left , only those to its  right ."

Danou · Answer

What do you mean by even values? Also why is the assumption that the "positive" integer for x and y is only 2 digits (and not bigger)? Help is appreciated

chetan2u · Answer

Even values meant even integers between 240 and 276.

Next if the numbers are 768964 and 876543865, we are looking for TENS digit of 2(768964 + 876543865). TENS digit will be given only by ones digit and tens digit, so why waste time on thinking what other digits are => 2(64+65)

Posted from my mobile device

KarishmaB · Answer

x and y both are of the form ...60 or ...61 or ... or ...69

When we add them (x + y), the sum ranges from
...60 + ...60 = ...20
...60 + ...61 = ...21
to
...68 + ...69 = ...37
...69 + ...69 = ...38

Last two digits will be anything from 20 to 38. 
When we multiply this sum by 2 to get 2(x+y), the value ranges from

2 * ...20 = ...40
2* ...21 = ...42
to
2*...37 = ...74
2*...38 = ...76

Hence the tens digit of 2(x+y) can be anything from 4 to 7 i.e. it can take 4 distinct values.

Answer (B)

anjanaroy · Answer

In both the cases when we multiply both X and Y individually 2 , the tens digit of 2X and 2Y can either be 2 or 3. Which means that the sum of tens digits of 2X and 2Y can be 4 (2+2) or 5 (2+3 , 3+2) or 6 (3+3).

Now on top of this just consider the maximum and minimum values the unit digit of (2X+2Y) can be which can be any even number between 0 to 16 (units of 2X or 2Y can be 0,2,4,6,8) . As long as the sum total of the unit digits is among 0,2,4,6,8 nothing needs to be added to the sum of the tens digit and it stays among 4 , 5 ,6 but when the sum total of unit digit is among 10, 12,14 ,16 then an additional 1 needs to be added to the tens digit and after adding 1 the tens digit can be 5(4+1) or 6(5+1) or 7(6+1) . So the probable values of tens digit can be  4  or  5  or  6  or  7  so a total of  Four  values and that's the answer

anjanaroy · Answer

In both the cases when we multiply both X and Y individually 2 , the tens digit of 2X and 2Y can either be 2 or 3. Which means that the sum of tens digits of 2X and 2Y can be 4 (2+2) or 5 (2+3 , 3+2) or 6 (3+3).

Now on top of this just consider the maximum and minimum values the unit digit of (2X+2Y) can be which can be any even number between 0 to 16 (units of 2X or 2Y can be 0,2,4,6,8) . As long as the sum total of the unit digits is among 0,2,4,6,8 nothing needs to be added to the sum of the tens digit and it stays among 4 , 5 ,6 but when the sum total of unit digit is among 10, 12,14 ,16 then an additional 1 needs to be added to the tens digit and after adding 1 the tens digit can be 5(4+1) or 6(5+1) or 7(6+1) . So the probable values of tens digit can be  4  or  5  or  6  or  7  so a total of  Four  values and that's the answer

monarchme · Answer

but nowhere its given that the x and y are two digit integers. so why we are assuming that ? it could be some x=abcd6e also

anjanaroy · Answer

It doesn't matter how many digits of integer they are because we are concerned about Tens digit so we would consider the digits only till tens digit from right which is the unit digit and tens digit . The tens digit when you multiply 261 with 2 and the tens digit when you multiply 234561 with 2 , in both e the case the tens digit is going to be 2. You only have to focus on 61 from the right . I hope this helps

Eshitadhar · Answer

Since the both nos have tens digital of 6.so when we add them it can two value 2 (when there is no carryover from unit digit) and 3 (when there is carryover from unit digit).now when this 2 at unit is multiplied by 2 it can again take 2 values (4 when there is no carryover from unit digit and 5 when thereis carryover from unit digit).same goes with 3( 6 and 7 is the value it can take).so total 4 values can be of tens digit of the given expression.

muskang2000 · Answer

x=6b; y=6d
so x=60+b, y=60+d

2(60+b)+2(60+d)
240+2(b+d)

Now, b and d both are single digits which can take any value from 0 to 9. Whats the max value of b+d? It's 18 (when b and d both are 9)

so, 240+2(18) = 276 so ten's digit is 7
next, 240+2(17)=274 so ten's digit is 7
you keep going (do mentally if you can) and you'll find:
276, 268, 258 and 248 as the unique numbers for different values of b + d. So, 7,6,5,4 are the possible ten's digits

AndyRugger · Answer

After studying the GMAT for 6 months, I've come to realize that questions like this punish most people who overly rely on formulas and trick knowledge (including myself.) A few people can think their way through it based on their grasp of number properties - and that's may be the fastest method - but the GMAT also rewards people who just crank on out of curiosity/desperation.

I rephased the 2x+2y answer as 2(x+y). So, what is the set of the possible values of the tens digit of this integer? No clue! - so let's go fishing.

Let x and y both equal 60. Tens digit of answer is a 2 (6+6=12).
I realize, trying out a few combinations of units digit, that I can only "carry over" a maximum of 1 from the units to the tens place, and therefore the tens digit of x+y either must be a 2 or a 3.
However, I'm trying to find 2(x+y). Not sure how that changes things - so again, let's go fishing.

I realize that the lowest possible x+y is 120 (60+60) and the greatest possible is 138 (69+69). No crazy number properties; it's just checking min/max at 0 and 9. Therefore, the least 2(x+y) is 240, and the greatest is 276. Doesn't take a genius to infer that this is the range of 2x+2y: 240-276. That means any number between 240 and 276 could be a solution, which means the tens digit could be 4, 5, 6, or 7.

Answer B: Four.

I realize this is difficult to do under time pressure on a test, but in the latter half of my studying, I've also realized that it's very difficult to formulate a planned solution path for many GMAT quant questions after first reading the stem. They are intentionally confusing and intimidating. Much of the time, you can get the right answer if you're willing to take the stem at face value, put pen to paper, and see where the numbers lead you.

If the tens digit of the positive integer x and the tens digit of the

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

If the tens digit of the positive integer x and the tens digit of the

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