Divisibility Test of 4

When the given numbers are small enough, say 2-digits or 3-digit numbers, it’s quite easy to check their divisibility by simple integers straight away through long division. But as the numbers get larger, it is often a hassle to perform all those long divisions. This is when these simple divisibility tests come to the rescue, which not only save plenty of time but also give us accurate results.

The Modulo Operation

Often in computing, in this case, divisibility and its related problems, we use an operation called the Modulo operation. This operation upon Euclidean division returns the remainder of a division after one number is divided by another. To put it in simpler words, in basic Euclidean division –

AB = Q Remainder R where,

A Dividend

B Divisor

Q Quotient

R Remainder

In order to obtain this remainder upon requirement, the modulo operation is used as –

A mod (B) = R, 

For example, 

– upon dividing 15 by 2, we get 152 = 7 remainder 1, this is written in modulo operation as, 

15 mod(2) = 1

– upon dividing 14 by 7, we get 147 = 2 remainder 0, this is written in modulo operation as, 

14 mod(7) = 0

Congruence Modulo

The congruence modulo is a congruence relation between 2 numbers, used when 2 numbers have the same remainder when divided by ‘n’. That is, for some integer k, 

a + b = kn  or  a – b = kn   for two numbers a and b

This is written as, a ≡ b (mod n) 

For example,

26 ≡ 11 (mod 5)  – this is so since 26 mod(5) and 11 mod(5) leave the same remainder

Divisibility Test by 4

In simple words, a number is said to be divisible by 4 if the number formed by its last two digits is divisible by 4 or the last two digits must be 0.

Taking the help of modulo operations, 

• Firstly, we see that from 100 onwards, that is, 102 onwards, all powers of 10 are divisible by 4   as  100 = 25×4, 1000 = 250×4, and so on….

• This can be generalized as – 10n ≡  0 (mod 4) for all  n>1

So, to check the divisibility of an integer N, say 2736, we write it as –

2736 ≡ (2×1000) + (7×100) + 36

2736 ≡ (2×0) + (7×0) + 36 (mod 4)

2736 ≡ 36 (mod 4)

Since 36 ≡ 0(mod 4),  hence 2736 ≡ 0(mod 4), proving that 2736 is divisible by 4

We can observe that all multiples of 4 are even, that is, all multiples of 4 are multiples of 2 but not vice-versa.

So simply, a given number is divisible by 4, if its last two digits as a number is divisible by 4

Alternative Test

We can also follow an alternative rule to find out if a given number is divisible by 4. This can be done by dividing the number by 2, and again checking if the quotient is divisible by 2, i.e, –

N divided by 2 gives quotient Q1 say, and if Q1 is divisible by 2, then the given number is divisible by 4 as a whole

For example, 

Take N = 1720 to check divisibility with 4

1720 2 = 860 = Q1

860 2 = 430   Hence since Q1  is divisible by 2, 1720 hence proved to be divisible by 4

Examples:

1. Let’s check the divisibility of 14540 by 4

• Following the general rule, taking the last digits of the given number as – “40”

• The number “40” is divisible by 4 as 4 x 10 = 40, and hence given number is divisible by 4

• Following the alternative rule, 145402 = 7270

and 7270 is divisible by 2 since it’s an even number

Thus, the given number is divisible by 4

2. Let’s check the divisibility of 7850 by 4

• Following the general rule, taking the last digits of the given number as – “50”

• The number “50” is not divisible by 4 as 50 mod(4) = 2, and hence given number is not divisible by 4

• Following the alternative rule, 78502 = 3925

and 3925 is not divisible by 2 since it’s an odd number

Thus, the given number is not divisible by 4

3. Let’s check the divisibility of 1432 by 4

• Following the general rule, taking the last digits of the given number as – “32”

• The number “32” is divisible by 4 as 4×8 = 32, and hence given number is divisible by 4

• Following the alternative rule, 14322 = 716

and 716 is divisible by 2 since it’s an even number

Thus, the given number is divisible by 4

4. Let’s check the divisibility of 80742 by 4

• Following the general rule, taking the last digits of the given number as – “42”

• The number “42” is not divisible by 4 as 42 mod(4) = 2, and hence given number is not divisible by 4

• Following the alternative rule, 807422 = 40371

and 40371 is not divisible by 2 since it’s an odd number

Thus, the given number is not divisible by 4

5. Let’s check the divisibility of 1,03,128 by 4

• Following the general rule, taking the last digits of the given number as – “28”

• The number “28” is divisible by 4 as 4×7 = 28, and hence given number is divisible by 4

• Following the alternative rule, 1,03,1282 = 51,564

and 51,564 is divisible by 2 since it’s an even number

Thus, the given number is divisible by 4

Conclusion

This article has given a clear concept of the divisibility test of 4 through proper explanations and examples. Hope this article has provided the students with all the insights that need in-depth understanding.