Let A be the largest positive integer that divides all the numbers of the form 3k + 4k + 5k and B be the largest positive integer that divides all the numbers of the form 4k + 3(4k) + 4k+2, where k is any positive integer. Then (A + B) equalsCorrect answer is '82'. Can you explain this answer?

Question

Answer

To find the largest positive integer that divides all the numbers of a specific form, we need to find the greatest common divisor (GCD) of those numbers. Let's find the GCD for each set of numbers separately.

Finding the GCD of numbers of the form 3k, 4k, and 5k:
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The GCD of any two numbers is the largest positive integer that divides both of them. So, we need to find the GCD of 3k and 4k, the GCD of 3k and 5k, and the GCD of 4k and 5k.

GCD of 3k and 4k:
The numbers 3k and 4k have a common factor of k. So, the GCD is k.

GCD of 3k and 5k:
The numbers 3k and 5k have no common factors other than k. So, the GCD is k.

GCD of 4k and 5k:
The numbers 4k and 5k have no common factors other than k. So, the GCD is k.

Since k is the GCD of all these pairs, it is also the GCD of the set {3k, 4k, 5k}. Therefore, A = k.

Finding the GCD of numbers of the form 4k, 3(4k), and 4k^2:
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Again, we need to find the GCD of each pair of numbers.

GCD of 4k and 3(4k):
The numbers 4k and 3(4k) have a common factor of 4k. So, the GCD is 4k.

GCD of 4k and 4k^2:
The numbers 4k and 4k^2 have a common factor of 4k. So, the GCD is 4k.

Since 4k is the GCD of both pairs, it is also the GCD of the set {4k, 3(4k), 4k^2}. Therefore, B = 4k.

Calculating the value of A + B:
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We need to find the value of A + B, which is k + 4k = 5k.

Since k can take any positive integer value, we need to find the largest possible value of 5k that satisfies the conditions. It is evident that k = 16 gives the largest value of 5k that is less than or equal to 100. Therefore, A + B = 5(16) = 80.

Hence, the correct answer is 80.

Answer

A is the HCF of 3^k + 4^k+ 5^kfor different values of k.

For k = 1, value is 12

For k = 2, value is 50

For k = 3, value is 216

HCF is 2. Therefore, A = 2

HCF of the values is when k = 1, i.e. 5*16 = 80

Therefore, B = 80

A + B = 82

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