there are only two numbers with the property that every digit except zero appears just once in the number and its square.one of these numbers is 854, since 854*854= 729316. the other number is made up of three consecutive digits in increasing order.what is the other number?
To find the other number, let's consider the properties mentioned:
1) Every digit except zero appears just once in the number
2) The number and its square have this property
First, we can list all the numbers with three consecutive digits in increasing order: 123, 234, 345, 456, 567, 678, 789.
Next, we can square these numbers to see if they satisfy property 1:
- 123*123 = 15129 (Missing 4, 6, 8)
- 234*234 = 54756 (Missing 1, 8, 9)
- 345*345 = 119025 (Missing 6, 7, 8)
- 456*456 = 207936 (Missing 1, 2, 5 and 7)
- 567*567 = 321489 (Missing 0, 2, 6)
- 678*678 = 459684 (Missing 1, 2, 3 and 9)
- 789*789 = 622521 (Missing 0, 3, 4 and 6)
Out of all of these, the only number that satisfies property 1 is 345. Therefore, the other number with the given properties is 345.
To find the other number, we can use a systematic approach.
First, let's analyze the number 854 and its square 729316.
For a number to meet the given property, each digit (except zero) should appear only once both in the number itself and its square.
In the number 854, we have three distinct digits: 8, 5, and 4. However, in the square (729316), we have an additional digit 9. This means that the number 854 does not meet the property requirement, as 9 appears in the square but not in the original number.
Now, we need to find a three-digit number that meets the property requirement. The number must have three consecutive digits, implying that the digits are in increasing order.
Let's list all possible three-digit numbers with increasing digits:
123
234
345
456
567
678
789
Next, we need to check if any of these numbers satisfy the property by finding the square of each number and checking if it contains all unique digits (except zero).
Let's calculate the squares of these numbers and check for uniqueness:
123 * 123 = 15129 (contains 1, 2, 5, 9)
234 * 234 = 54756 (contains 3, 4, 5, 6, 7)
345 * 345 = 119025 (contains 1, 2, 3, 5, 9)
456 * 456 = 207936 (contains 2, 3, 5, 6, 7, 9)
567 * 567 = 321489 (contains 1, 2, 3, 4, 6, 8, 9)
678 * 678 = 460284 (contains 0, 2, 4, 6, 8)
789 * 789 = 622521 (contains 2, 4, 6, 9)
After checking all the possible numbers, we can see that none of them satisfy the property requirement. Therefore, there is no three-digit number with consecutive increasing digits that fulfills the given property.
Hence, there is no other number that meets the given property.
Let's call the other number made up of three consecutive digits in increasing order as ABC.
To find the value of ABC, we can use the given property that every digit except zero appears just once in the number and its square.
Since the square of ABC should have unique digits except zero, we can start by finding the possible digits for each position:
For the units digit (C):
Since ABC is a three-digit number, its square will have at least 4 digits. Hence, the square's units digit cannot be equal to C. Therefore, we can eliminate C as a possible digit for the units place.
For the tens digit (B):
Since the square of a two-digit number will have at least three digits, the square's tens digit cannot be equal to B. Therefore, we can eliminate B as a possible digit for the tens place as well.
For the hundreds digit (A):
We can check the squares of the integers in increasing order until we find a square that satisfies the property. Since A is the hundreds digit, it can only take values from 1 to 6 since 729,316 has six digits.
By checking the squares of the numbers from 100 to 166, we find that 121 satisfies the given property (121*121 = 14,641). Therefore, the value of A is 1.
Now, we have ABC = 121.