[GMAT math practice 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
A palindrome, such as 12321, is a number that remains the sa
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- Max@Math Revolution
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Max@Math Revolution wrote:[GMAT math practice 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
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 76512, 311,244 and 2128 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: 2112
Likewise, if the last two digits are 36, then the first two digits are 63 to get the palindrome: 6336
And, if the last two digits are 80, then the first two digits are 08 to get the palindrome: 0880
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
- Max@Math Revolution
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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
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
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