The remainder when 23²³ is divided by 53 is

• (A) 17
• (B) 21
• (C) 30
• (D) 43

To find the remainder when \(23^{23}\) is divided by 53, we can use Fermat's Little Theorem. This theorem states that if \(p\) is a prime number and \(a\) is an integer not divisible by \(p\), then \(a^{p-1} \equiv 1 \mod p\).

Applying Fermat's Theorem

Here, \(p = 53\) and \(a = 23\). Since 23 is not divisible by 53, we can apply the theorem:

• Calculate \(p - 1\): \(53 - 1 = 52\).
• According to Fermat's theorem: \(23^{52} \equiv 1 \mod 53\).

Reducing the Exponent

Next, we need to reduce the exponent 23 modulo 52:

• Since \(23 < 52\), we can use \(23\) directly.

Calculating \(23^{23} \mod 53\)

Now we compute \(23^{23} \mod 53\). This can be simplified using successive squaring:

• Calculate \(23^1 \mod 53 = 23\).
• Calculate \(23^2 \mod 53 = 529 \mod 53 = 50\).
• Calculate \(23^4 \mod 53 = 50^2 \mod 53 = 2500 \mod 53 = 14\).
• Calculate \(23^8 \mod 53 = 14^2 \mod 53 = 196 \mod 53 = 37\).
• Calculate \(23^{16} \mod 53 = 37^2 \mod 53 = 1369 \mod 53 = 46\).

Combining Powers

Now, we combine the results to find \(23^{23}\):

• Since \(23 = 16 + 4 + 2 + 1\), we have:
• \(23^{23} \equiv 23^{16} \cdot 23^4 \cdot 23^2 \cdot 23^1 \mod 53\).
• Calculating this gives:
• \(46 \cdot 14 \cdot 50 \cdot 23 \mod 53\).

Final Calculation

Calculating step-by-step:

• First, \(46 \cdot 14 = 644 \mod 53 = 8\).
• Next, \(8 \cdot 50 = 400 \mod 53 = 25\).
• Finally, \(25 \cdot 23 = 575 \mod 53 = 31\).

Thus, the remainder when \(23^{23}\) is divided by 53 is 31, which is not listed in the options provided. Please check the options again or the problem statement for any discrepancies.