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This mock test of Test: Unit Digit- 1 for UPSC helps you for every UPSC entrance exam.
This contains 15 Multiple Choice Questions for UPSC Test: Unit Digit- 1 (mcq) to study with solutions a complete question bank.
The solved questions answers in this Test: Unit Digit- 1 quiz give you a good mix of easy questions and tough questions. UPSC
students definitely take this Test: Unit Digit- 1 exercise for a better result in the exam. You can find other Test: Unit Digit- 1 extra questions,
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QUESTION: 1

What will be the units digit, when the product of the first 10 natural numbers is divided by 100?

Solution:

QUESTION: 2

Find the product of 57 x 61 x 39 x 53.

Solution:

QUESTION: 3

Find the units digit of the product of all the prime numbers between 1 and 13^{13}.

Solution:

QUESTION: 4

What is the rightmost non-zero digit of 90^{42}?

Solution:

QUESTION: 5

Find the units digit of 5^{3n} + 9^{5m}, where m and n are positive integers

- m is an odd integer
- n is an even integer

Solution:

QUESTION: 6

If x = 3^{21} and y = 6^{55}, what is the remainder when xy is divided by 5?

Solution:

QUESTION: 7

What is the units digit of 31467^{32 }× 97645^{23} × (32168^{5} + 8652)^{479}?

Solution:

**Step 1: Question statement and Inferences**

We need to find the units digit of all the terms in the expression 31467^{32 }× 97645^{23} × (32168^{5} + 8652)^{479}

**Step 2: Finding required values**

**Given: **

We Know,

2 → 2, 4, 8, 6 (Corresponding powers: 4m +1,4m+2, 4m +3, 4m)

5 → 5: Every power will have the same last digit.

7 → 7, 9, 3, 1 (Corresponding powers: 4m +1,4m+2, 4m +3, 4m)

8 → 8, 4, 2, 6 (Corresponding powers: 4m +1,4m+2, 4m +3, 4m)

Considering the first term: 31467^{32}

The unit digit of this term will be determined by 7^{32}

32 = 4*8

--> 32 is of the form 4m

--> 7’s power is of the form 4m

--> The units digit of 7^{32 }will be 1

--> The units digit of 31467^{32} is 1

Considering the second term: 97645^{23}

The units digit of this term will be determined by 5^{23}

The units digit of 5 raised to the power anything is 5.

--> The units digit of 5^{23 }is 5

--> The units digit of 97645^{23 }is 5

Considering the third term: 32168^{5}

The units digit of this term will be determined by 8^{5}

5 = 4*1 +1

--> 5 is of the form 4m +1

--> 8’s power is of the form 4m +1

--> The units digit of 8^{5 }will be 8

--> The units digit of 32168^{5} is 8

The units digit of the last term, 8652, is 2

**Step 3: Calculating the final answer**

Units digit of 31467^{32} × 97645^{23} × (32168^{5} + 8652)^{479 }= 1 × 5 (8 + 2)^{479} = 1 × 5 (0)^{479} = 1 × 5 × 0 = 0

**Answer: Option (A) **

QUESTION: 8

Find the rightmost non-zero digit of the number 345637300^{3725}.

Solution:

**Step 1: Question statement and Inferences**

We are given the number 345637300^{3725}, and we have to find the rightmost non-zero digit of this number. We know that the rightmost digit is the unit digit of a number.

Now, the expression can be written as follows:

345637300^{3725} = (345673 * 100)^{3725}

= 345673^{3725} * 100^{3725}

Now, we know that the rightmost non-zero digit of the number will come from the expression 345673^{3725}.

Also, the unit digit of the expression 345673^{3725} = the unit digit of 3^{3725}

Thus, we have to find the unit digit of 3^{3725}.

**Step 2: Finding required values **

We know that every 4th power of 3 has the same unit digit and cycles of power of 3 are 3, 9, 7, and 1.

3^{4m + 1} = 3

3^{4m + 2} = 9

3^{4m + 3} = 7

3^{4m }= 1

Now, 3725 = 3700 + 25

= 4*k + 4*6 + 1 (Since every number which is a multiple of 100 is a multiple of 4)

So, 3725 = 4m + 1, where m is some positive integer

Thus, the unit digit of 3^{3725} = the unit digit of 3^{4m + 1} = 3

**Step 3: Calculating the final answer**

So, the rightmost non-zero digit of the number 345637300^{3725} = 3.

**Answer: Option (B)**

QUESTION: 9

If p is a positive integer, what is the units digit of Z, if Z = (104^{4p + 1}) * (277^{p + 1}) * (93^{p + 2}) * (309^{6p}) ?

Solution:

**Step 1: Question statement and Inferences**

We are given that Z = (104^{4p + 1}) * (277^{p + 1}) * (93^{p + 2}) * (309^{6p}). We have to find the units digit of Z.

Here we can say that:

The unit digit of Z = The units digit of the product of the unit digits of the given numbers

Now, we also know that the unit digit of any power of a number depends only on the unit digit of the number. Thus, we can write the expression as:

Z = (4^{4p + 1}) * (7^{p + 1}) * (3^{p + 2}) * (9^{6p})

**Step 2: Finding required values **

Z = (4^{4p + 1}) * (7^{p + 1}) * (3^{p + 2}) * (9^{6p})

Next, let’s find the unit digit of the individual expressions:

Unit digit of 4^{4p + 1}:

Every second power of 4 has the same unit digit.

Cycles of powers of 4 are 4, 6, 4, 6 …

So, unit digit of 4^{4p + 1} = 4 (Since 4p + 1 is an odd number and every odd power of 4 has the unit digit as 4)

Unit digit of 9^{6p}:

Every second power of 9 has the same unit digit.

Cycles of powers of 9 are 9, 1, 9, 1 …

So, unit digit of 9^{6p} = 1 (Since 6p is an even number and every even power of 9 has the unit digit 1)

Now, the cyclicity of the numbers 3 and 7 is 4. So, we can’t decide the unit digit of the expression 3^{p+2} and 7^{p+1} since we don’t know the value of p. However, the product of these numbers can be further solved as follows:

7^{p + 1} * 3^{p + 2} = 7^{p + 1} * 3^{p + 1} * 3

= (7*3)^{p + 1} * (3) (Since a^{m} * b^{m} = (ab)^{m} )

= (21)^{p + 1} * (3)

Now, we know that the unit digit of the expression 21^{p+1} will always be 1 since any power of 1 always gives a unit digit 1.

Thus, the unit digit of (21)^{p + 1} * (3) = 1 * 3 = 3

**Step 3: Calculating the final answer**

Now, let’s plug in all the values in the expression for Z.

Z = 4 * 3 * 1 = 12

So, the unit digit of Z will be 2.

**Answer: Option (A)**

QUESTION: 10

If p and q are positive integers and X = 6^{p} + 7^{q+23}, what is the units digit of X?

(1) q = 2p – 11

(2) q^{2} – 10q + 9 = 0

Solution:

__Steps 1 & 2: Understand Question and Draw Inferences__

We are given that X = 6^{p} + 7^{q+23}, and we have to find the unit digit of X. The numbers p and q both are positive integers. Now, the unit digit of X will be the sum of the unit digits of 6^{p} and 7^{q+23}.

So, we have to find the unit digit of 6^{p} and 7^{q+23}.

Now, we know that the unit digit of 6 raised to any integer power is 6. So, the unit digit of 6^{p} is 6.

And the cyclicity of 7 is 4 i.e. every 4th power of 7 has the same unit digit and cycles of power of 7 are 7, 9, 3, and 1.

So, the unit digit of the expression:

X = 6^{p} + 7^{q+23} = (Unit Digit of 6^{p}) + (Unit digit of 7^{q+23})

= 6 + (Unit digit of 7^{q+23})

So, the unit digit of 7^{q+23} will depend on the value of q. We have to find the value of q to determine the unit digit of X.

__Step 3: Analyze Statement 1__

Statement 1 says: q = 2p – 11

However, since we don’t know the value of p, we can’t determine the value of q.

Hence, statement I is not sufficient to answer the question: What is the unit digit of X?

__Step 4: Analyze Statement 2__

Statement 2 says:

q^{2} – 10q + 9 = 0

q^{2} – 9q -q + 9 = 0

(q – 9) (q – 1) = 0

Thus, q= 1, 9

Let’s consider both the values one by one:

- If q = 1

In this case, the expression 7^{q+23} becomes:

7^{1+23 }= 7^{24}

Since the cyclicity of 7 is 4 i.e. every 4th power of 7 has the same unit digit and cycles of power of 7 are 7, 9, 3, and 1.

And,

24 = 4*6

So, the unit digit of 7^{24} = 1

2. If q = 9

In this case, the expression 7^{q+23} becomes:

79^{+23 }= 7^{32}

Since the cyclicity of 7 is 4 i.e. every 4th power of 7 has the same unit digit and cycles of power of 7 are 7, 9, 3, and 1.

And,

32 = 4*8

So, the unit digit of 7^{32} = 1

So, in both the cases we get the unit digit of 7^{q+23} as 1. As derived in the first step, the unit digit of X

= 6 + (Unit digit of 7^{q+23}) = 6 + 1 = 7

So, statement (2) alone is sufficient to answer the question: What is the unit digit of X?

__Step 5: Analyze Both Statements Together (if needed)__

Since statement (2) alone is sufficient to answer the question, we don’t need to perform this step.

**Answer: Option (B)**

QUESTION: 11

If the number 653 *xy* is divisible by 90, then (*x* + *y*) = ?

Solution:

90 = 10 x 9

Clearly, 653*xy* is divisible by 10, so *y* = 0

Now, 653*x*0 is divisible by 9.

So, (6 + 5 + 3 + *x* + 0) = (14 + *x*) is divisible by 9. So, *x* = 4.

Hence, (*x* + *y*) = (4 + 0) = 4.

QUESTION: 12

3897 x 999 = ?

Solution:

3897 x 999= 3897 x (1000 - 1)

= 3897 x 1000 - 3897 x 1

= 3897000 - 3897

= 3893103.

QUESTION: 13

What is the unit digit in 7^{105} ?

Solution:

Unit digit in 7^{105} = Unit digit in [ (7^{4})^{26} x 7 ]

But, unit digit in (7^{4})^{26} = 1

Unit digit in 7^{105} = (1 x 7) = 7

QUESTION: 14

Which of the following numbers will completely divide (4^{61} + 4^{62} + 4^{63} + 4^{64}) ?

Solution:

(4^{61} + 4^{62} + 4^{63} + 4^{64}) = 4^{61} x (1 + 4 + 4^{2} + 4^{3}) = 4^{61} x 85

= 4^{60} x (4 x 85)

= (4^{60} x 340), which is divisible by 10.

QUESTION: 15

106 x 106 - 94 x 94 = ?

Solution:

106 x 106 - 94 x 94= (106)^{2} - (94)^{2}

= (106 + 94)(106 - 94)

[**Ref:** *(a ^{2} - b^{2}) = (a + b)(a - b)*]

= (200 x 12)

= 2400.

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