Key Concepts and Formulas Used

The solution to this problem relies on fundamental principles of number theory and divisibility:

1. Divisibility Rule for $a^n - b^n$: For any integers $a$ and $b$, and any positive integer $n$, the expression $a^n - b^n$ is always divisible by $a-b$.
   • Extension (when $n$ is even): If $n$ is an even positive integer, then $a^n - b^n$ is also divisible by $a+b$.
2. Modular Arithmetic: This is a system of arithmetic for integers, which considers the remainder of a number when it is divided by a fixed positive integer (called the modulus). We write $a \equiv b \pmod{m}$ if $a$ and $b$ have the same remainder when divided by $m$, or equivalently, if $a-b$ is divisible by $m$.
3. Fermat's Little Theorem: If $p$ is a prime number, then for any integer $a$ not divisible by $p$, we have $a^{p-1} \equiv 1 \pmod{p}$. This is useful for simplifying large exponents in modular arithmetic.

---

Step-by-Step Solution

We are asked to determine the divisibility of the expression $E = 25^{190}-19^{190}-8^{190}+2^{190}$ by 14 and 34.

To check divisibility by composite numbers, we must check divisibility by their prime factors:

• Divisibility by $14 \implies$ Divisibility by 2 AND Divisibility by 7.
• Divisibility by $34 \implies$ Divisibility by 2 AND Divisibility by 17.

---

1. Check Divisibility by 2

• Reasoning: An integer is divisible by 2 if it is an even number. We can determine the parity (odd or even) of each term.
• $25^{190}$: The base 25 is odd. An odd number raised to any positive integer power is always odd. So, $25^{190}$ is Odd.
• $19^{190}$: The base 19 is odd. An odd number raised to any positive integer power is always odd. So, $19^{190}$ is Odd.
• $8^{190}$: The base 8 is even. An even number raised to any positive integer power (greater than 0) is always even. So, $8^{190}$ is Even.
• $2^{190}$: The base 2 is even. An even number raised to any positive integer power is always even. So, $2^{190}$ is Even.

Now, substitute these parities into the expression: $E = \text{Odd} - \text{Odd} - \text{Even} + \text{Even}$ We know that:

• $\text{Odd} - \text{Odd} = \text{Even}$

• $\text{Even} - \text{Even} = \text{Even}$

• Therefore, $E = \text{Even} - \text{Even} = \text{Even}$.

• Conclusion: The expression $E$ is an even number, and thus divisible by 2.

---

2. Check Divisibility by 17

• Reasoning: We will use the divisibility rule $a^n - b^n$ is divisible by $a-b$. We will strategically group the terms to utilize this property with $a-b=17$.
• Rearrange the terms of the expression $E$ as follows: $E = (25^{190} - 8^{190}) - (19^{190} - 2^{190})$
• First Group: Consider the term $25^{190} - 8^{190}$.
  • This is in the form $a^n - b^n$ with $a=25$, $b=8$, and $n=190$.
  • The difference $a-b = 25-8 = 17$.
  • According to the divisibility rule, $25^{190} - 8^{190}$ is divisible by $17$.
• Second Group: Consider the term $19^{190} - 2^{190}$.
  • This is in the form $a^n - b^n$ with $a=19$, $b=2$, and $n=190$.
  • The difference $a-b = 19-2 = 17$.
  • According to the divisibility rule, $19^{190} - 2^{190}$ is divisible by $17$.
• Combining Results: Since both $(25^{190} - 8^{190})$ and $(19^{190} - 2^{190})$ are divisible by 17, their difference must also be divisible by 17. $E = (\text{Multiple of } 17) - (\text{Multiple of } 17) = \text{Multiple of } 17$
• Conclusion: The expression $E$ is divisible by 17.

---

3. Check Divisibility by 7

• Reasoning: We will use modular arithmetic to find the remainder when $E$ is divided by 7. If the remainder is 0, it's divisible by 7; otherwise, it's not.
• First, find the remainder of each base when divided by 7:
  • $25 \equiv 4 \pmod{7}$
  • $19 \equiv 5 \pmod{7}$
  • $8 \equiv 1 \pmod{7}$
  • $2 \equiv 2 \pmod{7}$
• Substitute these congruences into the original expression $E$: $E \equiv 4^{190} - 5^{190} - 1^{190} + 2^{190} \pmod{7}$
• Simplify terms:
  • $1^{190} = 1$.
  • Notice that $5 \equiv -2 \pmod{7}$. Since the exponent 190 is an even number, $(-2)^{190} = 2^{190}$. So, $5^{190} \equiv (-2)^{190} \equiv 2^{190} \pmod{7}$.
• Substitute these simplifications back into the congruence for $E$: $E \equiv 4^{190} - 2^{190} - 1 + 2^{190} \pmod{7}$ $E \equiv 4^{190} - 1 \pmod{7}$
• Now, we need to evaluate $4^{190} \pmod{7}$. Let's find the cycle of powers of $4 \pmod{7}$:
  • $4^1 \equiv 4 \pmod{7}$
  • $4^2 \equiv 16 \equiv 2 \pmod{7}$
  • $4^3 \equiv 4 \times 2 \equiv 8 \equiv 1 \pmod{7}$ The cycle length for powers of $4 \pmod{7}$ is 3, as $4^3 \equiv 1 \pmod{7}$.
• Divide the exponent 190 by the cycle length 3:
  • $190 = 3 \times 63 + 1$
• Therefore, $4^{190} = 4^{3 \times 63 + 1} = (4^3)^{63} \times 4^1$.
• Applying modular arithmetic: $4^{190} \equiv (1)^{63} \times 4 \equiv 1 \times 4 \equiv 4 \pmod{7}$.
• Substitute this result back into the congruence for $E$: $E \equiv 4 - 1 \pmod{7}$ $E \equiv 3 \pmod{7}$
• Conclusion: Since the remainder is 3 (not 0), the expression $E$ is not divisible by 7.

---

4. Final Determination of Divisibility by 14 and 34

Based on our detailed analysis:

• Divisibility by 14: For $E$ to be divisible by 14, it must be divisible by both 2 and 7.

  • We found $E$ is divisible by 2.
  • We found $E$ is NOT divisible by 7.
  • Therefore, $E$ is not divisible by 14.

• Divisibility by 34: For $E$ to be divisible by 34, it must be divisible by both 2 and 17.

  • We found $E$ is divisible by 2.
  • We found $E$ is divisible by 17.
  • Therefore, $E$ is divisible by 34.

Combining these results, the expression $25^{190}-19^{190}-8^{190}+2^{190}$ is divisible by 34 but not by 14. This corresponds to option (D).

---

Tips and Common Mistakes to Avoid

• Decomposition for Composite Divisors: Always break down composite divisors (like 14 and 34) into their prime factors (2, 7, and 2, 17, respectively) and check divisibility by each prime factor individually. A number must be divisible by all prime factors to be divisible by the composite number.
• Strategic Grouping: For expressions involving multiple terms with powers, look for ways to group terms to apply divisibility rules like $a^n \pm b^n$ by $a \pm b$.
• Modular Arithmetic for Large Exponents: When dealing with large exponents, modular arithmetic is indispensable. Remember to find the cycle length of powers modulo a number (e.g., $a^k \equiv 1 \pmod{m}$) to simplify calculations. Fermat's Little Theorem is a powerful shortcut for prime moduli.
• Parity in Modular Arithmetic: Be mindful of negative bases in modular arithmetic. For an even exponent $n$, $(-x)^n = x^n$. For an odd exponent $n$, $(-x)^n = -x^n$.

---

Summary and Key Takeaway

This problem elegantly combines divisibility rules and modular arithmetic. The key takeaway is to systematically break down the problem into checking divisibility by prime factors (2, 7, and 17) and applying the appropriate techniques: parity checks for 2, the $a^n - b^n$ rule for 17, and modular exponentiation for 7. Careful calculation with modular arithmetic, especially with large exponents, is crucial to arriving at the correct conclusion. Based on our calculations, the expression is divisible by 34 but not by 14.