Number System

The greatest five-digit number is 99999. Divide 99999 by 1246: 99999 / 1246 = 80.25 Thelargest integer quotient is 80. Multiply the divisor by this quotient: 1246 * 80 = 99680. Alternatively, find the remainder: 99999 = 1246 * 80 + remainder i.e, 99999 = 99680 + remainder =)remainder = 99999 - 99680 = 319. Thegreatest5-digitmultipleis99999 - Remainder = 99999 - 319 = 99680. This matches option (1).

To find the remainder when is divided by 7, we can use Fermat's Little Theorem, which states that if is a prime and is an integer not divisible by , then .

Here, and . According to Fermat's Little Theorem:

Thus, the exponent 2000 can be reduced modulo 6:

equals 2, because:

with a remainder of 2.

Therefore, .

Calculating :

.

Finding :

with a remainder of .

Hence, .

The remainder when is divided by 7 is therefore 1.