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655-705 Level|   Math Related|         
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Anugmat740
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GMAT Club Olympics 2024 (Day 10): A palindrome is a number that reads 20 Jul 2024, 04:36
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Smallest- 111
Largest - 393
Min=1, Max=9
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GMAT Club Olympics 2024 (Day 10): A palindrome is a number that reads 20 Jul 2024, 04:42
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­all such nos will be 101, 111, 121.... 191 then 909...919......999
when yu will multiply these nos, units didgits will be sum of central digit, ex in for 101*101 it will be 0 +0 & for 181*181 it will be 9,
Product of nos starting with 9 & ending will behave differently

so 0 will be lowest & 9 being highest
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GMAT Club Olympics 2024 (Day 10): A palindrome is a number that reads 20 Jul 2024, 04:53
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0 is not possible as product has 1 as the unit digit
minimum: consider- 919 ..product=81..minimum is 1
maximum: consider- 393..product=81..maximum is 9

Answer 1 & 9­
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GMAT Club Olympics 2024 (Day 10): A palindrome is a number that reads 20 Jul 2024, 05:32
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# is a 3 digit palindrome
Product of 3 digits ends with 1
We can eliminate all even digits and 5
The only digits we can use are 1,3,7,9

The numbers that satisfy all four criteria: 797, 919, 393

Max value of tens digit = 9
Min value of tens digit = 1
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GMAT Club Olympics 2024 (Day 10): A palindrome is a number that reads 20 Jul 2024, 06:33
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Bunuel
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.

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GMAT Club Olympics 2024 (Day 10): A palindrome is a number that reads 20 Jul 2024, 06:41
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Lowest - 111 , maximum - 797

Posted from my mobile device
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GMAT Club Olympics 2024 (Day 10): A palindrome is a number that reads 20 Jul 2024, 06:46
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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.

3-digit palindromes whose product of digits has 1 as the unit digit: 111, 393, 797
Minimum 1: => 111
Maximum: 9 => 797­
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GMAT Club Olympics 2024 (Day 10): A palindrome is a number that reads 20 Jul 2024, 06:53
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Three Digit palindrome number can be of the form xxx or xyx

Now x^3 last digit =1
or x{^2}*y last digit =1
We need to find minimum x or y
Minimum x=1, when Number = 111

Maximum y=9 when Number 393

Imo 1-2, 2-6
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GMAT Club Olympics 2024 (Day 10): A palindrome is a number that reads 20 Jul 2024, 07:00
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To solve the problem, we need to find all 3-digit palindromes such that the product of their digits has 1 as the unit digit.

A 3-digit palindrome can be represented as aba , where a and b are digits, and a≠0 because it's a 3-digit number.
The product of the digits in the palindrome is a×b×a = a^2 ×b.

For the product
a^2 ×b to have 1 as the unit digit, we need to find suitable values of a and b.
Let's analyze:
The unit digit of a^2 must multiply with b to give a unit digit of 1.

Considering possible unit digits:
If a ends in 1: a^2 ends in 1.
If a ends in 2: a^2 ends in 4.
If a ends in 3: a^2 ends in 9.
If a ends in 4: a^2 ends in 6.
If a ends in 5: a^2 ends in 5.
If a ends in 6: a^2 ends in 6.
If a ends in 7: a^2 ends in 9.
If a ends in 8: a^2 ends in 4.
If a ends in 9: a^2 ends in 1.
If a ends in 0: a^2 ends in 0 (not possible since a≠0).
We see that a ending in 1 or 9 will make
a2 end in 1, which is necessary for the product to end in 1 when multiplied by b.

Next, we need b such that
1×b ends in 1. The only digit
b can be is 1 because 1×1=1.

So, we only need to consider palindromes of the form a1a .

The palindromes fitting the criteria are:

If a=1: the number is 111.
If a=9: the number is 919.
So, the minimum possible value of the tens digit of any of these numbers is 1, and the maximum possible value is 1.

Therefore, the answers are:

Minimum: 1
Maximum: 1­
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GMAT Club Olympics 2024 (Day 10): A palindrome is a number that reads 20 Jul 2024, 07:22
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Catman
Three Digit palindrome number can be of the form xxx or xyx

Now x^3 last digit =1
or x{^2}*y last digit =1
We need to find minimum x or y
Minimum x=1, when Number = 111

Maximum y=9 when Number 393

Imo 1-2, 2-6
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­Bunuel I have not received Kudos for this explanation.Can you please check? Thanks.­
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GMAT Club Olympics 2024 (Day 10): A palindrome is a number that reads 20 Jul 2024, 07:32
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Catman

Catman
Three Digit palindrome number can be of the form xxx or xyx

Now x^3 last digit =1
or x{^2}*y last digit =1
We need to find minimum x or y
Minimum x=1, when Number = 111

Maximum y=9 when Number 393

Imo 1-2, 2-6
­Bunuel I have not received Kudos for this explanation.Can you please check? Thanks.­
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