Question:medium

Let \(A\) be the largest positive integer that divides all the numbers of form \(3^k+4^k+5^k\), and \(B\) be the largest positive integer that divides all the numbers of the form \(4^k+3(4^k)+4^{k+2}\), where k is any positive integer. Then \((A+B)\) equals

Updated On: Jan 15, 2026
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Solution and Explanation

Step 1: Evaluating \( 3^k + 4^k + 5^k \)

Calculate the expression for various \( k \) values:

  • For \( k = 1 \): \( 3^1 + 4^1 + 5^1 = 3 + 4 + 5 = 12 \)
  • For \( k = 2 \): \( 3^2 + 4^2 + 5^2 = 9 + 16 + 25 = 50 \)
  • For \( k = 3 \): \( 3^3 + 4^3 + 5^3 = 27 + 64 + 125 = 216 \)

Determine the Highest Common Factor (HCF) of these results: \[ \gcd(12, 50, 216) = 2 \] Thus, \( A = 2 \)

Step 2: Evaluating \( 4^k + 3(4^k) + 4^{k+2} \)

Simplify the expression: \[ 4^k + 3(4^k) + 4^{k+2} = 4^k(1 + 3) + 4^{k+2} = 4^{k+1} + 4^{k+2} = 4^{k+1}(1 + 4) = 5 \cdot 4^{k+1} \]

Calculate the expression for various \( k \) values:

  • For \( k = 1 \): \( B = 5 \cdot 4^{2} = 5 \cdot 16 = 80 \)
  • For \( k = 2 \): \( B = 5 \cdot 4^{3} = 5 \cdot 64 = 320 \)
  • For \( k = 3 \): \( B = 5 \cdot 4^{4} = 5 \cdot 256 = 1280 \)

Determine the HCF: \[ \gcd(80, 320, 1280) = 80 \] Thus, \( B = 80 \)

Final Step: Sum of A and B

\[ A + B = 2 + 80 = \boxed{82} \]

Answer:

\( \boxed{82} \)

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The passage given below is followed by four summaries. Choose the option that best captures the essence of the passage.
In the dynamic realm of creativity, artists often find themselves at the crossroads between drawing inspiration from diverse cultures and inadvertently crossing into the territory of cultural appropriation. Inspiration is the lifeblood of creativity, driving artists to create works that resonate across borders. In a globalized era of the modern world, artists draw from a vast array of cultural influences. When approached respectfully, inspiration becomes a bridge, fostering understanding and appreciation of cultural diversity. However, the line between inspiration and cultural appropriation can be thin and easily blurred.
Cultural appropriation occurs when elements from a particular culture are borrowed without proper understanding, respect, or acknowledgment. This leads to the commodification of sacred symbols, the reinforcement of stereotypes, and the erasure of the cultural context from which these elements originated. It is essential to recognize that the impact of cultural appropriation extends beyond the realm of artistic expression, influencing societal perceptions and perpetuating power imbalances.
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