If 6X2 is a three digit number and it is also divisible by 7 and 8 . Find what is the value of x which is the middle number of the three digit number?

Question

If 6x2 is a three digit number and it is also divisible by 7 and 8. Find, what is the value of x in the middle of the three digit number ?

Answer 1

The number can be expressed 600+10X+2=602+10X.

This number is divisible by 7 and 8, which means it's divisible by 56, so:

602+10X=56n where n is a whole number.

This is the same as 10X=56n-602, so X=(60n-4n-600-2)/10=6n-4n/10-60-2/10.

This can be written: X=(6n-60)-(4n+2)/10=6n-60-(2n+1)/5.

6n-60 is a whole (positive) number because n is a whole number.

When (2n+1)/5 is a whole number we can see that we get a whole number when n=2, n=7, n=12, etc.

6n-60 has to be positive, so n≥10. For example, n=12 gives us 72-60=12, from which we need to subtract (2×12+1)/5=5, giving us 12-5=7. We can expect X=7 to be the missing digit.

Let's test it: 672/7=96; 672/8=84. So 672 is the number.