Consider four-digit numbers for which the first two digits are equal and the last two digits are also equal. How many are such numbers perfect squares? (2007)
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Consider four-digit numbers for which the first two digits are equal and the last two digits are also equal. How many are such numbers perfect squares?
a. 2
b. 4
c. 0
d. 1 [CAT, 2007]
Solution:
Consider four digit-numbers
Numerals and Digits:
The first part of this question says: Consider four digits numbers whose first and last two digits are equal
Let’s assume that the four digits unknown number is in the form of AABB.
Since the first and the second digits are equal, this is why we have chosen AA and, for the same reason BB for the third and the fourth digits because they are equal.
Perfect Square: [Note: Perfect Square means the number multiplied by itself i.e., 2 x 2 = 4; 3 x 3 = 9 and so on].
Since digits can range from 1 to 9 in this case. (0 can’t be considered here). If 1 is chosen for the first digit and 2 for the second digit, then the following forms emerge:
1122 Where A=1 and B=2
2233 Where A=2 and b=3
3344 Where A=3 and B=4
4455 Where A=4 and B=5
5555 is wrong because the first two and the last two digits are not equal in this case.
This means the next digits are 5566, 6677, 7788 and 8899.
Now, If we divide any of the above numerals by 11.
(Note – all the numerals above (1122, 2233…8899), it can be seen, are divisible by 11 only)
We get \frac { 1122 }{ 11 } =102 (this form) where A=1 and B=2.
So, all the numerals can be written now as (A0B) x 11 where (A0B) is a perfect square. (according to the question)
Now, we have to write all the perfect squares, which exist in the number system, starting from the lowest 4 (2 x 2) till 81 (9 x 9) and multiply by 11 to find which of the products take (s) A0B form.
▪ 11 x 4= 44, 11 x 25= 275, 11 x 64= 704,
▪ 11 x 9= 99, 11 x 36 = 426, 11 x 81= 891
▪ 11 x 16= 176, 11 x 49 = 539
Since only one product takes A0B form when A = 7 and B = 4, the numeral is 7744, there is only one square in AoB form. The answer is only 1 square.