The total number of four-digit numbers such t | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Let the four-digit number be $abcd$,where $a \in \{1, 2, \dots, 9\}$ and $b, c, d \in \{0, 1, \dots, 9\}$. Since $a, b, c$ are divisible by $d$,$d

Vedclass · Accepted Answer

$1086$

Vedclass · Answer

(D) Let the four-digit number be $abcd$,where $a \in \{1, 2, \dots, 9\}$ and $b, c, d \in \{0, 1, \dots, 9\}$. Since $a, b, c$ are divisible by $d$,$d$ cannot be $0$. For each $d \in \{1, 2, \dots, 9\}$,we count the possible values for $a, b, c$: $d=1$: $a \in \{1, \dots, 9\}$ ($9$ choices),$b, c \in \{0, \dots, 9\}$ ($10$ choices each). Total: $9 	imes 10 	imes 10 = 900$. $d=2$: $a \in \{2, 4, 6, 8\}$ ($4$ choices),$b, c \in \{0, 2, 4, 6, 8\}$ ($5$ choices each). Total: $4 	imes 5 	imes 5 = 100$. $d=3$: $a \in \{3, 6, 9\}$ ($3$ choices),$b, c \in \{0, 3, 6, 9\}$ ($4$ choices each). Total: $3 	imes 4 	imes 4 = 48$. $d=4$: $a \in \{4, 8\}$ ($2$ choices),$b, c \in \{0, 4, 8\}$ ($3$ choices each). Total: $2 	imes 3 	imes 3 = 18$. $d=5$: $a \in \{5\}$ ($1$ choice),$b, c \in \{0, 5\}$ ($2$ choices each). Total: $1 	imes 2 	imes 2 = 4$. $d=6$: $a \in \{6\}$ ($1$ choice),$b, c \in \{0, 6\}$ ($2$ choices each). Total: $1 	imes 2 	imes 2 = 4$. $d=7$: $a \in \{7\}$ ($1$ choice),$b, c \in \{0, 7\}$ ($2$ choices each). Total: $1 	imes 2 	imes 2 = 4$. $d=8$: $a \in \{8\}$ ($1$ choice),$b, c \in \{0, 8\}$ ($2$ choices each). Total: $1 	imes 2 	imes 2 = 4$. $d=9$: $a \in \{9\}$ ($1$ choice),$b, c \in \{0, 9\}$ ($2$ choices each). Total: $1 	imes 2 	imes 2 = 4$. Summing these: $900 + 100 + 48 + 18 + 4 + 4 + 4 + 4 + 4 = 1086$.

The total number of four-digit numbers such that each of the first three digits is divisible by the last digit is equal to:

Similar Questions

If $^{n}P_{4} = 24 \times ^{n}C_{5}$,then the value of $n$ is

Assuming that the balls of the same color are identical,find the number of ways to select one or more balls from $10$ white,$9$ green,and $7$ black balls.

How many different words can be formed by jumbling the letters in the word $MISSISSIPPI$ in which no two $S$ are adjacent?

What is the remainder when $1! + 2! + 3! + \dots + 200!$ is divided by $14$?

The number of words that do not start and end with vowels,formed using all the letters of the word $'UNIVERSITY'$ such that all vowels are in alphabetical order,is

Vedclass Test Series

Exam Paper Generator

Online Exam Module