Problem: Find out whether 1634829⁸³⁶⁵² + 6738213⁹²⁶³⁴ is divisible by 4.

Solution: Now, at first this problem seems out of reach for any human mind in the world due to the sheer immensity of this number, but trust me, its actually not that difficult once you make some observations.

First, let me clarify a bit of terminology. note that if two integers a and b are said to be congruent to each other modulo p, it means that a and b leave the same remainder when divided by p. For example, 8 is congruent to 3 modulo 5 because both 8 and 3 leave a remainder of 3 when divided by 5.

Result 1: The sum of two numbers is congruent to the sum of their remainders when divided by p modulo p. Proof: Let the two numbers be a and b, and let their remainders be c and d respectively. Let c + d = kp + r (It leaves a remainder of r modulo p). a + b = (ep + c) + (fp + d) = (e + f + k)p + r. So a + b also leaves a remainder of r modulo p, as desired.

Result 2: The product of two numbers is congruent to the product of their remainders when divided by p modulo p. Proof: Let the two numbers be a and b, and let their remainders be c and d respectively. Let cd = kp + r (It leaves a remainder of r modulo p). ab = (ep + c)(fp + d) = (efp + ed + cf + k)p + r. So ab also leaves a remainder of r modulo p, as desired.

Now note that both the exponents in the question are even whereas both bases are odd. An odd number either leaves a remainder of 1 or 3 modulo 4. However, an odd square is congruent to 1*1 = 1 or 3*3 = 9 = 1 modulo 4 (Using Result 2). So an odd square always leaves a remainder of 1 modulo 4. Both terms in the question are an odd number raised to the power of an even number, which can be re-written as an odd square raised to an integer power (as an even number is 2 * an integer).

So, using result 2, the first term is congruent to 1 ^ (83652/2) and the second term is congruent to 1 ^ (92634/2), which are both equal to 1 (Recall that 83652 and 92634 were the exponents in the question). Using result 1, I know that their sum leaves a remainder of 1 + 1 = 2, so the expression in the problem leaves a remainder of 2 when divided by 4. Hence, the answer is that 1634829⁸³⁶⁵² + 6738213⁹²⁶³⁴ is not divisible by 4!!

More generally, any number of the form odd^even + odd^even is not divisible by 4. This can easily be seen from the solution, as the only property of the numbers in the question which was utilised by us was whether they are odd or even.

Don’t forget to check my blog next week for another fascinating problem!!