divisibilty test?

Question

if x-5 is divisible by 5
if x-6 is divisible by 6

can we say x is divisible by 5 , 6 ?

if yes , how ? example

Answer 1

Yes we can

x-5 = 5*i
x-6 = 6*j

That means x = 5*(i+1)
x = 6*(j+1)
Thus x is a multiple of both 5 and 6.

Answer 2

If x - 5 is divisible by 5,
then (x - 5) / 5 = A,
where A is an integer.

But (x - 5)/ 5 = x /5 - 1 = A, and so x/5 = A + 1.
A + 1 is certainly an integer, which shows x is divisible by 5.

Similarly, if (x - 6) / 6 = B, then x/6 = B + 1,
for another integer B, so x must be divisible by 6.

If x is divisible by 5 and 6, then it must be divisible by 30,
so the only values for x are multiples of 30, e.g. 60, 90, 120,...

Answer 3

Yes
General proof for any integer n

if you divide x-n by n you obtain
(x-n)/n = x/n -n/n = x/n -1
(this expression is an integer if and only if x/n is an integer)

ex in your case take the number 120
120-5 =115 and 115/5 =23
120-6 =114 and 114/6 =19

Answer 4

Yes, yes we can.
Think of the basic multiplication scenarios.
If you have 15 baskets that have six apples each, taking one basket away, you still have a number which is divisible by 6.
If you take those original 90 apples, and a few more baskets, you could have 18 baskets with 5 apples in each. Thus you can take one away, and have 17 apples with 5 apples in each. This way, we know that our original number (90) is divisible by both 5 and 6.

Answer 5

YES!

We have a rule that if we have an expression where a divides (b+c) and b, then it must also divide c
Since 5 divides x-5 and also divides 5, it must also divide x
Since 6 divides x-6 and also divides 6, it must also divide x

Since 5 and 6 are relatively prime (no same factors) then x must be divisible by 5 and 6

=]

EXAMPLE:
x=30
30-5 = 25, div by 5
30 - 6 = 24, div by 6
30 is div by 5 and 6