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19 Jul 2024, 10:03
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Bunuel
3 digit palindrome number = \(xyx\)
x and y vary from 1-9 (0 can not be possible as multiplication of all 3 digits will be 0)
Multiplication = \(x^2*y\)
As, unit place of (unit place of x^2 * unit place of y) = unit place of multiplication of x^2 and y
As, x is from 1-9, x^2 can be any of 1, 4, 9, 16, 25, 36, 49, 64, 81.
Here with \(x^2 = 1, 81\) and \(y = 1\) we can get the unit place as 1 and Yminimum = 1
Also with \(x^2 = 9, 49\) and\( y = 9\) we can get the unit place as 1 and Ymaximum = 9
19 Jul 2024, 10:10
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A palindrome is a number that reads the same when read from either end. For example, 12321 is a palindrome. A set consists of all the 3-digit palindromes whose product of digits has 1 as the unit digit. Select for Minimum the minimum possible value of the tens digit of any of the numbers in the set, and select for Maximum the maximum possible value of the tens digit of any of the numbers in the set. Make only two selections, one in each column.
For minimum possible value of tens digit:
0 wont work as multiplication would be 0.
Considering 1, We can think of number 919. It fits the criteria of unit digit as 1.
For maximum possible value of tens digit:
Considering 9, We can think of number 393. It fits the criteria of unit digit as 1.
Hence, 1 and 9.
For minimum possible value of tens digit:
0 wont work as multiplication would be 0.
Considering 1, We can think of number 919. It fits the criteria of unit digit as 1.
For maximum possible value of tens digit:
Considering 9, We can think of number 393. It fits the criteria of unit digit as 1.
Hence, 1 and 9.
19 Jul 2024, 10:24
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Bunuel
The only way to get an odd number when you look at the product is odd*odd.
The product of the digits aba (since we're only looking at 3 digit palindromes), is \(a^2*b\). That means that \(a^2\) must be odd (so a must be odd) and b must be odd as well.
If we are looking for the min, we can test what happens if \(b=1\) and \(a^2=1\). These multiplied will give us 1 so it works. That means that the smallest possible value of the tens digit is 1 (note that 0 will give us a product of 0).
To get the max, we can test \(b=9\). Now we must find a way to get the units digit of the product to 1.
If \(a=1\), we will get a units digit of 9 -> not valid
If \(a=3\), we will get a units digit of 1 -> valid
The max possible digit in the tens is 9.
The answer is:
Minimum: 1
Maximum: 9
