Question

How many four-digits numbers, each divisible by 4 can be formed using the digits 1, 2, 3, 4 and 5, repetition of digits being allowed in any number?
(a) 100
(b) 150
(c) 125
(d) 75

Answer 1

Hint: First write divisibility rule of 4, check the digits possible with application of rule of 4. Write the cases one by one with usage of the rule in each case. By adding all possible cases you will get the total numbers of 4-digit numbers divisible by 4. This total is the required result in the question given.

Complete step-by-step answer:
Given digits in the question can be written in form of 1, 2, 3, 4, 5.
Given condition in the question can be written in the form of a 4- digit number divisible by 4.
Divisibility by 4: The number may have any digits but if the last two digits are divisible by 4 then the whole number is divisible by 4.
Possibilities for the last two digits of the required 4-digit number. If the first digit is assumed to be as I, we get a 2-digit number as: 12.
If the first digit is assumed to be 2, we get a 2-digit number as: 24.
If first digit is assumed to be as 3, we get 2-digit number as: 32
If the first digit is assumed to be 4, we get a 2-digit number as: 44 (as repetition is allowed).
If first digit is assumed to be as 5, we get 2-digit number as: 52
Now these all 2 digits numbers are the last two digits of your required 4-digit numbers, So, we can write them.
As repetition is allowed you can repeat the use of digits in the last two places also.
For every case the possibility remains the same as they are following the same condition.
The number of possible second digits of our required number 5.
The possibilities of first digit of our requirements: 5
Rule of sum: In combination, the rule of sum or addition principle is basic counting principle. It is simply defined as, if there are A, ways of doing P work and B ways of doing Q work. Given P, Q works cannot be done together. Total number of ways to do both P, Q are given by (A+B) ways.
Rule of product: In combination, the rule of product or multiplication principle is basic counting principle. It is simply defined as, if these are A ways of doing P work and B ways of doing Q work. Given P, Q works can be done at a time. Total number of ways to do both P, Q works are given by $ (A\times B) $ ways.
By above two definitions, we select to use product rule. Total number of possibilities, we can write it in the form of $ 5\times 5 $ .
By simplifying the above, we can write its value to be: 25.
This is a possibility for one case. There are 5 such cases namely the last two digits being 12, 24, 32, 44, 52. So, total number of cases for numbers, can be written as: \[25+25+25+25+25=25\times 5\]
By simplifying the above, we get its value to be:
Total cases= 125
So, the correct answer is “Option C”.

Note: Generally, students forget the condition of repetition and forget to consider the case with 44. So, always consider all cases. Students also confuse and use sum rules instead of product rules. So, look at definitions carefully before using them. Because using an incorrect rule may lead to a case where the whole answer is wrong.