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What is the greatest possible common divisor of two different positive
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01 Jul 2021, 08:03
01 Jul 2021, 08:03
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What is the greatest possible common divisor of two different positive integers which are less than 144?
A. 143
B. 142
C. 72
D. 71
E. 12
A. 143
B. 142
C. 72
D. 71
E. 12
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Re: What is the greatest possible common divisor of two different positive
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01 Jul 2021, 08:05
01 Jul 2021, 08:05
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Carcass wrote:
What is the greatest possible common divisor of two different positive integers which are less than 144?
A. 143
B. 142
C. 72
D. 71
E. 12
A. 143
B. 142
C. 72
D. 71
E. 12
The key word here is different
If the two numbers were allowed to be the same, then we could use 143 and 143, in which case, the GCD = 143
However, since the two numbers must be DIFFERENT, then we might first try to do something with 143 and some other number.
Since 143 = (11)(13), we can see that, in order to MAXIMIZE the GCD of the two numbers, the other number must be 13
So, 143 and 13 have a GCD of 13
Now let's try 142 and some other number.
Since 142 = (2)(71), we can see that, in order to MAXIMIZE the GCD of the two numbers, the other number must be 71
So, 142 and 71 have a GCD of 71
Following this logic, we can see that 142 and 71 will MAXIMIZE the GCD of the 2 numbers.
So, the correct answer is D
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Re: What is the greatest possible common divisor of two different positive
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28 Oct 2022, 08:12
28 Oct 2022, 08:12
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We need to find What is the greatest possible common divisor(GCD) of two different positive integers which are less than 144
Let's solve it using two methods
Method 1: Logic
Now the numbers are less than 144
=> We can take the numbers as 142 and 143, but then their GCD = 1 as they will have only 1 as the common factor.
If we think about this then we will realize that we can take two numbers such that one number is a big number (lets say x) *1 and other is a big number (lets say x) * 2
=> \(\frac{144}{2}\) = 72 can be taken as x and the numbers can be taken as 72*1 = 72 and 72*2 = 144
But, the numbers have to be less than 144
=> We can try with \(\frac{142}{2}\) = 71 and 71*2 = 142
So, Answer will be D
Method 2: Eliminate Option Choices
A. 143 For GCD of two different numbers to be 143 both have to be multiple of 143
=> We can take the numbers as 143*1 and 143*2 = 286, but numbers CANNOT be ≥ 144 => NOT POSSIBLE
B. 142 For GCD of two different numbers to be 142 both have to be multiple of 142
=> We can take the numbers as 142*1 and 142*2 = 284, but numbers CANNOT be ≥ 144 => NOT POSSIBLE
C. 72 For GCD of two different numbers to be 72 both have to be multiple of 72
=> We can take the numbers as 72*1 and 72*2 = 144, but numbers CANNOT be ≥ 144 => NOT POSSIBLE
D. 71 For GCD of two different numbers to be 71 both have to be multiple of 71
=> We can take the numbers as 71*1 and 71*2 = 142, which are both < 144 => POSSIBLE
So, Answer will be D
Hope it helps!
To learn more about LCM and GCD watch the following videos
Let's solve it using two methods
Method 1: Logic
Now the numbers are less than 144
=> We can take the numbers as 142 and 143, but then their GCD = 1 as they will have only 1 as the common factor.
If we think about this then we will realize that we can take two numbers such that one number is a big number (lets say x) *1 and other is a big number (lets say x) * 2
=> \(\frac{144}{2}\) = 72 can be taken as x and the numbers can be taken as 72*1 = 72 and 72*2 = 144
But, the numbers have to be less than 144
=> We can try with \(\frac{142}{2}\) = 71 and 71*2 = 142
So, Answer will be D
Method 2: Eliminate Option Choices
A. 143 For GCD of two different numbers to be 143 both have to be multiple of 143
=> We can take the numbers as 143*1 and 143*2 = 286, but numbers CANNOT be ≥ 144 => NOT POSSIBLE
B. 142 For GCD of two different numbers to be 142 both have to be multiple of 142
=> We can take the numbers as 142*1 and 142*2 = 284, but numbers CANNOT be ≥ 144 => NOT POSSIBLE
C. 72 For GCD of two different numbers to be 72 both have to be multiple of 72
=> We can take the numbers as 72*1 and 72*2 = 144, but numbers CANNOT be ≥ 144 => NOT POSSIBLE
D. 71 For GCD of two different numbers to be 71 both have to be multiple of 71
=> We can take the numbers as 71*1 and 71*2 = 142, which are both < 144 => POSSIBLE
So, Answer will be D
Hope it helps!
To learn more about LCM and GCD watch the following videos
