Question

Find the number in the form x3579x that is divisible by 42.

Answer 1

Numerical values for the first "x" would include the integers 1-9. The "grunt" method here would be to divide 13579x by 42, then 23579x by 42, and so on, to determine whether or not a value of x can be found that results in NO remainder. Let's assume that "x" has the same value at the end of each "x3479x as it does in the beginning:

135791

235792

335793

and so on.

Note immediately that x must be an even digit, since no multiple of 42 results in an odd final digit to x3479x.

Thus, from x=1 through 9, we eliminate 1, 3, 5, 7, 9 and investigate x = 2, 4, 6, 8.

Does 42 divide into 235792 without a remainder? No.

Does 42 divide into 435794 without a remainder? No.

Does 42 divide into 635796 without a remainder? Yes. The quotient is 15138.

So: the desired number is 635796, and 42 / 635796 = 15138.

Answer 2

Haven't seen this one on Brainly yet.

x3579x = 100000 x + 35790 + x

100000 = 2380 × 42 + 40

35790 = 852×42 + 6

Working mod 42,

$100000 x + 35790 + x \equiv 40 x + 6 + x \equiv 41x + 6 \equiv 6-x \pmod{42}$

So basically we need 6-x to be a multiple of 42, so between 0 and 9 that's x=6.

So our number is

635796 = 15138 × 42