A perfect square number has four digits , none of which is zero. The digits from left to right have values that are :even, even, odd, even. Find the number. If the number is 8836. Tell why and how did you get it .
See, the square root of the smallest 4-digit number, 1000, is somewhere between 31 and 32. So the smallest 4-digit square has to be the square of 32 (because 31 squared is too small), which is 1024.
The largest 4-digit number is 9999, which is one less than the square of 100. It is therefore the square of 99, or 9801. This means that the answer is the square of a number between 32 and 99, inclusive of both.
The unit digit is even, so the number is even. Thus, it is the square of an even number. So it is the square of one out of 32, 34, 36, … , 96, 98.
The leftmost digit has to be 2, 4, 6 or 8. Let us narrow down our search to all squares between 2000 and 2998. Square root of 2000 is between 44 and 45, while that of 3000 is between 54 and 55. So the squares of numbers 46, 48, 50, 52, 54 are what count here. Check all five. Is the answer one amongst them?
- 46 - 2116 - X
- 48 - 2304 - X
- 50 - 2500 - X
- 52 - 2704 - X
- 54 - 2916 - X
Clearly, not one of these. The next range is 4000 to 4999, and the square root of 4000 is between 63 and 64. We start with 64. The square root of 5000 is between 70 and 71. We end at 70. The even numbers in this range are 64, 66, 68, 70.
Watch out for this one:
- 64 - 4096 - no, because it has a 0 in it
- and so on...
Eventually you get to the answer, which is:
8836
See, the square root of the smallest 4-digit number, 1000, is somewhere between 31 and 32. So the smallest 4-digit square has to be the square of 32 (because 31 squared is too small), which is 1024.
The largest 4-digit number is 9999, which is one less than the square of 100. It is therefore the square of 99, or 9801. This means that the answer is the square of a number between 32 and 99, inclusive of both.
The unit digit is even, so the number is even. Thus, it is the square of an even number. So it is the square of one out of 32, 34, 36, … , 96, 98.
The leftmost digit has to be 2, 4, 6 or 8. Let us narrow down our search to all squares between 2000 and 2998. Square root of 2000 is between 44 and 45, while that of 3000 is between 54 and 55. So the squares of numbers 46, 48, 50, 52, 54 are what count here. Check all five. Is the answer one amongst them?
- 46 - 2116 - X
- 48 - 2304 - X
- 50 - 2500 - X
- 52 - 2704 - X
- 54 - 2916 - X
Clearly, not one of these. The next range is 4000 to 4999, and the square root of 4000 is between 63 and 64. We start with 64. The square root of 5000 is between 70 and 71. We end at 70. The even numbers in this range are 64, 66, 68, 70.
Watch out for this one:
- 64 - 4096 - no, because it has a 0 in it
- and so on...
Eventually you get to the answer, which is:
8836
