Finding Unit Digit - MCQ Test - UPSC MCQ

Answer: c)7

Solution:
The given product is (7493²⁶³)x(151²⁹)
Required unit digit = the unit digit of(3²⁶³)x(1²⁹) ...(1)
In the value of 3 to the power 4, we have the unit digit as 1.
so, we can rewrite (3²⁶³) = [3^{(4x65 + 3)}] = [(3⁴)⁶⁵] x (3³)
Then from eqn(1),
The unit digit of(3²⁶³)x(1²⁹) = The unit digit of[(3⁴)⁶⁵] x (3³) x 1²⁹
= The unit digit of[1⁶⁵] x 27 x 1
= The unit digit of 1 x 7 x 1 = 7
Hence, the answer is 7.

Given that 634²⁶² + 634²⁶³
= 634²⁶²(1 + 634)
= (634²⁶²) x 635
The unit digit of (634²⁶²) x 635 = the unit digit of (4²⁶² x 5)
We know that, the unit digit of 4 to the power of any odd number is 4 and the unit digit of 4 to the power of any even number is 6.
Then the unit digit of (4²⁶² x 5) = unit digit of(6 x 5) = 0

We are tasked with finding the digit in the unit place of the number represented by 7⁹⁵ × 3⁵⁸. To do this, we need to examine the units digits of 7⁹⁵ and 3⁵⁸ separately, and then multiply them together.

1. Units digit of 7⁹⁵:

   The units digits of powers of 7 follow a repeating pattern:

   • 7¹ = 7 (units digit = 7)
   • 7² = 49 (units digit = 9)
   • 7³ = 343 (units digit = 3)
   • 7⁴ = 2401 (units digit = 1)

   The pattern repeats every 4 terms: 7, 9, 3, 1.

   Now, to find the units digit of 7⁹⁵, divide 95 by 4:

   95 ÷ 4 = 23 remainder 3. Thus, the units digit of 7⁹⁵ corresponds to the units digit of 7³, which is 3.

2. Units digit of 3⁵⁸:

   The units digits of powers of 3 follow a repeating pattern:

   • 3¹ = 3 (units digit = 3)
   • 3² = 9 (units digit = 9)
   • 3³ = 27 (units digit = 7)
   • 3⁴ = 81 (units digit = 1)

   The pattern repeats every 4 terms: 3, 9, 7, 1.

   Now, to find the units digit of 3⁵⁸, divide 58 by 4:

   58 ÷ 4 = 14 remainder 2. Thus, the units digit of 3⁵⁸ corresponds to the units digit of 3², which is 9.

3. Multiplying the units digits:

   Now, multiply the units digits of 7⁹⁵ and 3⁵⁸:

   • Units digit of 7⁹⁵ = 3
   • Units digit of 3⁵⁸ = 9

   Multiply 3 and 9:

   3 × 9 = 27, so the units digit of the product is 7.

Thus, the digit in the unit place of 7⁹⁵ × 3⁵⁸ is 7. Therefore, the correct answer is Option A: 7.

We are tasked with finding the units digit of 39⁶¹. To solve this, we will focus on the units digit of powers of 39.

1. The units digit of 39 is the same as the units digit of 9. So, we need to find the units digit of 9⁶¹.
2. The powers of 9 follow a repeating pattern for their units digits:
   • 9¹ = 9 (units digit = 9)
   • 9² = 81 (units digit = 1)
   • 9³ = 729 (units digit = 9)
   • 9⁴ = 6561 (units digit = 1)
3. We observe that the units digits alternate between 9 and 1 in a cycle of 2: 9, 1, 9, 1, ...
4. To determine the units digit of 9⁶¹, we note that since 61 is an odd number, the units digit corresponds to 9¹, which is 9.

Thus, the units digit of 39⁶¹ is 9. Therefore, the correct answer is Option A: 9.