A palindrome, such as 12321, is a number that remains the same when it

Question

A palindrome, such as  12321 , is a number that remains the same when its digits are reversed. How many  4 -digit palindromes are divisible by  4 ?

A. 10 
 
 B. 20 
 
 C. 25 
 
 D. 28 
 
 E. 30

GMATGuruNY · Accepted Answer

For an integer to be divisible by 4, its last two digits must constitute a multiple of 4.
Since  \frac{99}{4} ≈ 24  there are 24 multiples of 4 between 01 and 99, inclusive.
Of these 24 multiples of 4, four options -- 20, 40, 60, 80 -- will yield a palindrome with a thousands digit of 0 and thus are not viable:
0220
0440
0660
0880
Thus, the total number of viable options  = 24-4 = 20

B

nick1816 · Answer

The unit digit of palindrome number can be 2, 4, 6 and 8
We know that for a 4-digit number to be divisible by 4, it's last 2 digits must be divisible by 4.

Case 1- When unit digit of 4-digit number is 2 or 6
tens digit can take 5 values ( 1,3,5,7 or 9)

Case 2- When unit digit of 4-digit number is 4 or 8
tens digit can take 5 values ( 0,2,4,6 or 8)

Total number of 4-digit palindromes that are divisible by 4= 4*5=20

Woody19 · Answer

nick1816

Case 1- When unit digit of 4-digit number is 2 or 6
tens digit can take 5 values ( 1,3,5,6 or 9)

62 and 66 are not divisible by 4.

Please explain.

sorry for the silly question.

Woody.

nick1816 · Answer

That was a typo. Thanks for pointing it out.
I have edited the solution.

Woody19 · Answer

nick1816

Thanks makes sense now.

Shishou · Answer

This is kind of tricky but here is how to work it out. There are probably better ways of doing this than I have done. We need to establish 2 baselines here : for a number to be divisible by 4 its last 2 digits MUST be divisible by 4 .So we are talking of numbers with last digits 12,16,20,24.......96. So we need to form a 4-digit number that reads the same forwards as it does backwards. For ex. 2112. is a palindrome and it is also divisible by 4 because the last 2 digits are divisible by 4. From 12 to 96 there are 1+ (96-12)/4= 22 numbers that are divisible by 4 . However multiples of 4 like 20,40,60 and 80 can not form palindromes. Try to form a 4-digit palindrome that ends in 80. We get 0880 which is just 88. So the total number will then be 22-4 = 18 possible palindromes.However, we must not forget that numbers ending in 04 and 08 are also divisible by 4 and can form palindromes. We can get 2 palindromes from both like this : 4004 and 8008. Added to the 18 we can get, we will have 20 possible palindromes that are divisible by 4. My two cents

BrentGMATPrepNow · Answer

If a number is divisible by 4, then the number created by the last 2 digits is divisible by 4. 
For example, we know that 765 12 , 311,2 44  and 21 28  are divisible by 4 because  12 ,  44 , and  28  are divisible by 4

So, the last 2 digits of the 4-digit palindromes must be  00 ,  04 ,  08 ,  12 , . . . ,  92  or  96  
There are   25   such values

KEY CONCEPT: Once we've selected the LAST 2 digits, we automatically know what the FIRST 2 digits must be, since we need to create a palindrome. 
For example, if the last two digits are  12 , then the first two digits are 21 to get the palindrome: 21 12 
Likewise, if the last two digits are  36 , then the first two digits are 63 to get the palindrome: 63 36 
And, if the last two digits are  80 , then the first two digits are 08 to get the palindrome: 08 80 
 STOP 
0880 is NOT a 4-digit number. 
In fact, any time the units digit is ZERO, the thousands digit of the 4-digit number is ZERO), which means we DON'T have a 4-digit number. 
So, we must SUBTRACT from   25   all of those instances in which units digit is ZERO

Let's count all of those instances:  00 ,  20 ,  40 ,  60  and  80 
There are  5  such instances.

So, the TOTAL number of 4-digit palindromes divisible by 4 =   25   -  5  =  20

Answer: B

Cheers,
Brent

MathRevolution · Answer

=>

Palindromes between  1000  and  10,000  have the form  ABBA .

When we check divisibility by  4 , we need only check the divisibility of  BA  by  4 .

The values of  BA  which are divisible by  4  are  04, 08, 12, 16, 24, 28, 32, 36, 44, 48, 52, 56, 64, 68, 72, 76, 84, 88, 92 , and  96  (note that  20, 40, 60, 80  and  00  are not possible values of  BA  as they do not give rise to  4 -digit numbers  ABBA ). The possible values for  ABBA  are  4004, 8008, 2112, 6116, 4224, 8228, 2332, 6336, 4444, 8448, 2552, 6556, 4664, 8668, 2772, 6776, 4884, 8888, 2992  and  6996 .
Thus, there are  20   4 -digit palindromes.

Therefore, the answer is B. 
Answer: B

Kinshook · Answer

A palindrome, such as  12321 , is a number that remains the same when its digits are reversed. How many  4 -digit palindromes are divisible by  4 ?

If last 2 digits are divisible by 4, palindrome will be divisible by 4.

When the last digit is either 2 or 6
Tens digit may be 1,3,5,7,9
2*5 = 10 cases

When the last digit is either 4 or 8
Tens digit may be 0,2,4,6,8
2*5 = 10 cases

Total 10 + 10 = 20 cases

IMO B

djangobackend · Answer

only last 2 digits are concerned - 
and last digit is even, 0,2,4,6,8 
with 0 at last palindrome won't be giving 3 digit so gone 
now with 2 at end, only 12, 32, 52, 72, 92 will work = 5 vals 
for 4 at end, 04 ,24, 44, 64, 84 = 5 vals 
6 at end - 16, 36, 56, 76, 96 = 5 vals 
8 at end - 08, 28, 48, 68, 88 = 5 vals 
total 20 vals

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.

A palindrome, such as 12321, is a number that remains the same when it

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

A palindrome, such as 12321, is a number that remains the same when it

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