Math / Discrete

QuestionQuestion 11 How many different numbers less than 1000 can be formed with distinct digits using the digits $0,2,3$ , $4 \& 8$ such that the number is divisible by 4 and the number does not end with 0 ?
Marks:3.0 Negative Marks 1.0

Studdy Solution

STEP 1

1. We are using the digits $0, 2, 3, 4, 8$ .
2. The number must be less than 1000.
3. The digits in the number must be distinct.
4. The number must be divisible by 4.
5. The number cannot end with 0.

STEP 2

1. Identify the condition for divisibility by 4.
2. List possible two-digit endings for the number.
3. Count valid combinations for each possible ending.
4. Sum the valid combinations to find the total number of numbers.

STEP 3

Identify the condition for divisibility by 4:
A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

STEP 4

List possible two-digit endings for the number using the digits $0, 2, 3, 4, 8$ that are divisible by 4 and do not end with 0:
- $24$ - $32$ - $48$

STEP 5

Count valid combinations for each possible ending:
- For ending $24$ : - Remaining digits: $0, 3, 8$ - Possible numbers: $3! = 6$
- For ending $32$ : - Remaining digits: $0, 4, 8$ - Possible numbers: $3! = 6$
- For ending $48$ : - Remaining digits: $0, 2, 3$ - Possible numbers: $3! = 6$

STEP 6

Sum the valid combinations to find the total number of numbers:
$6 + 6 + 6 = 18$
The total number of different numbers that can be formed is:
$\boxed{18}$

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QuestionQuestion 11 How many different numbers less than 1000 can be formed with distinct digits using the digits $0,2,3$ , $4 \& 8$ such that the number is divisible by 4 and the number does not end with 0 ?
Marks:3.0 Negative Marks 1.0

Studdy Solution

STEP 1

1. We are using the digits $0, 2, 3, 4, 8$ .
2. The number must be less than 1000.
3. The digits in the number must be distinct.
4. The number must be divisible by 4.
5. The number cannot end with 0.

STEP 2

1. Identify the condition for divisibility by 4.
2. List possible two-digit endings for the number.
3. Count valid combinations for each possible ending.
4. Sum the valid combinations to find the total number of numbers.

STEP 3

Identify the condition for divisibility by 4:
A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

STEP 4

List possible two-digit endings for the number using the digits $0, 2, 3, 4, 8$ that are divisible by 4 and do not end with 0:
- $24$ - $32$ - $48$

STEP 5

Count valid combinations for each possible ending:
- For ending $24$ : - Remaining digits: $0, 3, 8$ - Possible numbers: $3! = 6$
- For ending $32$ : - Remaining digits: $0, 4, 8$ - Possible numbers: $3! = 6$
- For ending $48$ : - Remaining digits: $0, 2, 3$ - Possible numbers: $3! = 6$

STEP 6

Sum the valid combinations to find the total number of numbers:
$6 + 6 + 6 = 18$
The total number of different numbers that can be formed is:
$\boxed{18}$

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QuestionQuestion 11 How many different numbers less than 1000 can be formed with distinct digits using the digits 0,2,30,2,30,2,3, 4&84 \& 84&8 such that the number is divisible by 4 and the number does not end with 0 ?Marks:3.0 Negative Marks 1.0

Studdy Solution

STEP 1

1. We are using the digits 0,2,3,4,80, 2, 3, 4, 80,2,3,4,8.2. The number must be less than 1000.3. The digits in the number must be distinct.4. The number must be divisible by 4.5. The number cannot end with 0.

STEP 2

1. Identify the condition for divisibility by 4.2. List possible two-digit endings for the number.3. Count valid combinations for each possible ending.4. Sum the valid combinations to find the total number of numbers.

STEP 3

Identify the condition for divisibility by 4:A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

STEP 4

List possible two-digit endings for the number using the digits 0,2,3,4,80, 2, 3, 4, 80,2,3,4,8 that are divisible by 4 and do not end with 0:- 242424 - 323232 - 484848

STEP 5

STEP 6

Sum the valid combinations to find the total number of numbers:6+6+6=18 6 + 6 + 6 = 18 6+6+6=18 The total number of different numbers that can be formed is:18 \boxed{18} 18​

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QuestionQuestion 11 How many different numbers less than 1000 can be formed with distinct digits using the digits $0,2,3$ , $4 \& 8$ such that the number is divisible by 4 and the number does not end with 0 ?
Marks:3.0 Negative Marks 1.0

1. We are using the digits $0, 2, 3, 4, 8$ .
2. The number must be less than 1000.
3. The digits in the number must be distinct.
4. The number must be divisible by 4.
5. The number cannot end with 0.

1. Identify the condition for divisibility by 4.
2. List possible two-digit endings for the number.
3. Count valid combinations for each possible ending.
4. Sum the valid combinations to find the total number of numbers.

Identify the condition for divisibility by 4:
A number is divisible by 4 if the number formed by its last two digits is divisible by 4.

List possible two-digit endings for the number using the digits $0, 2, 3, 4, 8$ that are divisible by 4 and do not end with 0:
- $24$ - $32$ - $48$

Sum the valid combinations to find the total number of numbers:
$6 + 6 + 6 = 18$
The total number of different numbers that can be formed is:
$\boxed{18}$