Test: Unit Digit- 1

Step 1: Question statement and Inferences

We need to find the units digit of all the terms in the expression 31467³² × 97645²³ × (32168⁵ + 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: 31467³²

The unit digit of this term will be determined by 7³²

32 = 4*8

-->  32 is of the form 4m

-->  7’s power is of the form 4m

-->  The units digit of 7³² will be 1

-->  The units digit of 31467³² is 1

 Considering the second term: 97645²³

The units digit of this term will be determined by 5²³

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

-->  The units digit of 5²³ is 5

-->  The units digit of 97645²³ is 5

 Considering the third term: 32168⁵

The units digit of this term will be determined by 8⁵

5 = 4*1 +1

-->  5 is of the form 4m +1

-->  8’s power is of the form 4m +1

-->  The units digit of 8⁵ will be 8

-->  The units digit of 32168⁵ is 8

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

Step 3: Calculating the final answer

Units digit of 31467³² × 97645²³ × (32168⁵ + 8652)⁴⁷⁹ = 1 × 5 (8 + 2)⁴⁷⁹ = 1 × 5 (0)⁴⁷⁹  = 1 × 5 × 0 = 0

 Answer: Option (A)

Step 1: Question statement and Inferences

We are given the number 345637300³⁷²⁵, 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:

345637300³⁷²⁵ = (345673 * 100)³⁷²⁵

             = 345673³⁷²⁵ * 100³⁷²⁵

Now, we know that the rightmost non-zero digit of the number will come from the expression 345673³⁷²⁵.

Also, the unit digit of the expression 345673³⁷²⁵ = the unit digit of 3³⁷²⁵

Thus, we have to find the unit digit of 3³⁷²⁵.

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 3³⁷²⁵ = the unit digit of 3^{4m + 1} = 3

Step 3: Calculating the final answer

So, the rightmost non-zero digit of the number 345637300³⁷²⁵ = 3.

Answer: Option (B)

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)

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: 

q² – 10q + 9 = 0

 q² – 9q -q + 9 = 0 

(q – 9) (q – 1) = 0

 Thus, q= 1, 9

 Let’s consider both the values one by one:

1. If q = 1

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

7¹⁺²³ = 7²⁴

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 7²⁴ = 1

     2.  If q = 9

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

79⁺²³ = 7³²

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 7³² = 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)

90 = 10 x 9
Clearly, 653xy is divisible by 10, so y = 0
Now, 653x0 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.

3897 x 999= 3897 x (1000 - 1)
= 3897 x 1000 - 3897 x 1
= 3897000 - 3897
= 3893103.

Unit digit in 7¹⁰⁵ = Unit digit in [ (7⁴)²⁶ x 7 ]
But, unit digit in (7⁴)²⁶ = 1
 Unit digit in 7¹⁰⁵ = (1 x 7) = 7

(4⁶¹ + 4⁶² + 4⁶³ + 4⁶⁴) = 4⁶¹ x (1 + 4 + 4² + 4³) = 4⁶¹ x 85
= 4⁶⁰ x (4 x 85)
= (4⁶⁰ x 340), which is divisible by 10.

106 x 106 - 94 x 94= (106)² - (94)²
= (106 + 94)(106 - 94)
[Ref: (a² - b²) = (a + b)(a - b)]
= (200 x 12)
= 2400.