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How do i prove that 52563744 is divisible by 24 ?

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by regor60 » Wed May 04, 2016 9:59 am
OsMaker wrote:How do i prove that 52563744 is divisible by 24 ?
I'm sure there are some abstruse divisibility rules that could be applied here, but not knowing those, break the number up into two pieces:

52,560,000/24 + 3744/24

these individual divisions go quickly:

2,190,000 + 156

It's pretty clear from inspection that the original number is at least divisible by 4, but the necessary 3 is a little harder to see.
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by chetan.sharma » Wed May 04, 2016 2:42 pm
OsMaker wrote:How do i prove that 52563744 is divisible by 24 ?
Hi,
for being div by 24, the numbershould be div by 3 and 8..
for 8 or 2^3, we check for div of last three digits of number..
so if 744 is div by 8, the entire number will be div by 8..
744/8 = 93.. so YES
now 3..
for a number to be div by 3, the sum of its digits must be div by 3..
so 52563744 = 5+2+5+6+3+7+4+4 = 36..
36 is div by 3, so the number is also div by 3..

since 52563744 is div by both 3 and 8, the number is div by 24..

hope it helps
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by OptimusPrep » Wed May 04, 2016 7:52 pm
OsMaker wrote:How do i prove that 52563744 is divisible by 24 ?
Here is a bit more about the divisibility rules:

Divisibility of 2 - The last digit should be divisible by 2
Divisibility of 3 - Sum of the digits should be multiple of 3
Divisibility of 4 - Last two digits should be divisible by 4
Divisibility of 5 - Last digit should be 5 or 0
Divisibility of 6 - Number is even and if the sum of its digits is divisible by 3
Divisibility of 7 - Quiet complex and should not be used
Divisibility of 8 - Last three digits should be divisible by 8
Divisibility of 9 - Sum should be divisible by 9
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by OsMaker » Wed May 04, 2016 8:12 pm
chetan.sharma wrote:
OsMaker wrote:How do i prove that 52563744 is divisible by 24 ?
Hi,
for being div by 24, the numbershould be div by 3 and 8..
for 8 or 2^3, we check for div of last three digits of number..
so if 744 is div by 8, the entire number will be div by 8..
744/8 = 93.. so YES
now 3..
for a number to be div by 3, the sum of its digits must be div by 3..
so 52563744 = 5+2+5+6+3+7+4+4 = 36..
36 is div by 3, so the number is also div by 3..

since 52563744 is div by both 3 and 8, the number is div by 24..

hope it helps

Thanks your method is what i was looking for
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by OsMaker » Wed May 04, 2016 8:59 pm
chetan.sharma wrote:
OsMaker wrote:How do i prove that 52563744 is divisible by 24 ?
Hi,
for being div by 24, the numbershould be div by 3 and 8..
for 8 or 2^3, we check for div of last three digits of number..
so if 744 is div by 8, the entire number will be div by 8..
744/8 = 93.. so YES
now 3..
for a number to be div by 3, the sum of its digits must be div by 3..
so 52563744 = 5+2+5+6+3+7+4+4 = 36..
36 is div by 3, so the number is also div by 3..

since 52563744 is div by both 3 and 8, the number is div by 24..

hope it helps
There is one more thing you didn't mention:

24 should be broken down such that the number should be co-primes (in this case 3 and 8 are co-primes)

Consider the following problem:

Show that whether 537804 is divisible by 24.

Ans: Now 24 can be broken as 6 x 4, but since the 6 and 4 are not co-primes this won't work.

Let's test whether 537804 is divisible by 24 or not.

537804 is divisible by 6 because it obeys following conditions:

1. 537804 is even
2. Sum of digits of 537804 ( 5 + 3 + 7 + 8 + 0 + 4 ) is divisible by 3.

537804 is divisible by 4 because the last two digits is divisible by 4.

Therefore 537804 must be divisible by 24. But it is not.

This contradiction arises because 6 and 4 are not co-primes. Lets again break down 24 so that it's factors are co-primes

24 = 8 x 3, here 8 and 3 are co-primes.

537804 is not divisible by 8 so it is not divisible by 24.
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