Question

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

Answer 1

Step 1: Values of the form \( 3^k + 4^k + 5^k \)

Let’s compute for different values of \( k \):

• 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 \( k = 3 \): \( 3^3 + 4^3 + 5^3 = 27 + 64 + 125 = 216 \)

Now compute the HCF of these results: \[ \gcd(12, 50, 216) = 2 \] So, \( A = 2 \)

Step 2: Values of the form \( 4^k + 3(4^k) + 4^{k+2} \)

We simplify: \[ 4^k + 3(4^k) + 4^{k+2} = 4^k(1 + 3) + 4^{k+2} = 4^{k+1} + 4^{k+2} = 4^{k+1}(1 + 4) = 5 \cdot 4^{k+1} \]

Evaluate for different values of \( k \):

• 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 \( k = 3 \): \( B = 5 \cdot 4^{4} = 5 \cdot 256 = 1280 \)

Now compute the HCF: \[ \gcd(80, 320, 1280) = 80 \] So, \( B = 80 \)

Final Step: Sum of A and B

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

Answer:

\( \boxed{82} \)

Answer 2

Let \( A \) be the HCF of the expression \( 3^k + 4^k + 5^k \) for different values of \( k \).

Step-by-step Evaluation:

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 \( k = 3 \):
\( 3^3 + 4^3 + 5^3 = 27 + 64 + 125 = 216 \)

Now compute the HCF of \( 12, 50, 216 \):
\[ \gcd(12, 50, 216) = 2 \] Hence, \( A = 2 \).

Another Expression:

Consider this algebraic manipulation:
\( 4^k + 3(4^k) + 4^{k+2} \)

Simplifying step-by-step:

• \( 4^k + 3(4^k) = 4^k (1 + 3) = 4^k \times 4 = 4^{k+1} \)
• So full expression: \( 4^{k+1} + 4^{k+2} \)
• Factor out \( 4^{k+1} \):
  \( = 4^{k+1}(1 + 4) = 4^{k+1} \times 5 \)

When \( k = 1 \):
\[ B = 5 \times 4^{1+1} = 5 \times 4^2 = 5 \times 16 = 80 \]

Final Calculation:

Now combine both results:
\[ A + B = 2 + 80 = \boxed{82} \]

Answer:

\( \boxed{82} \)