Consider the arithmetic expression E = 18^{41} - 22^{60} \times 37^{42}. When the value of… — State PSC Exam

Consider the arithmetic expression $E = 18^{41} - 22^{60} \times 37^{42}$ . When the value of $E$ is divided by $10$ , what is the resulting positive remainder?

A
8
4Answer
C
6
D
9

Answer

Evaluating the remainders of each term modulo 10 (which is equivalent to finding their unit digits), we obtain 8, 6, and 9 respectively. Applying the correct order of operations, multiplication precedes subtraction:

8 - (6 \times 9) = 8 - 54 = -46

. To convert -46 into a valid positive remainder modulo 10, we add the nearest larger multiple of 10 (which is 50), yielding

-46 + 50 = 4

Step-by-Step Solution

Find the unit digit (remainder modulo 10) of

18^{41}

The unit digit cycle of 8 is (8, 4, 2, 6). Since 41 divided by 4 leaves a remainder of 1, the unit digit corresponds to the first value in the cycle, which is

8^1 = 8

Find the unit digit (remainder modulo 10) of

22^{60}

The unit digit cycle of 2 is (2, 4, 8, 6). Since 60 is perfectly divisible by 4 (remainder 0), we use the 4th value in the cycle, which is

2^4 = 16 \rightarrow 6

Find the unit digit (remainder modulo 10) of

37^{42}

The unit digit cycle of 7 is (7, 9, 3, 1). Since 42 divided by 4 leaves a remainder of 2, the unit digit corresponds to the second value in the cycle, which is

7^2 = 49 \rightarrow 9

Substitute these values into the expression following the order of operations.

8 - 6 \times 9 = 8 - 54 = -46

According to the BODMAS rule, multiplication must be performed before subtraction.

Convert the negative result to a valid positive remainder modulo 10.

To convert a negative remainder to a positive one, add a multiple of the divisor that brings the value above zero.

-46 + 50 = 4

Key Concept

Unit digit cyclicity, modular arithmetic, and the conversion of negative remainders.

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