[ equations in integers ] [ Decimal number system ]

a) In a three-digit number, the first digit from the left was erased, then the resulting two-digit number was multiplied by 7 and the original three-digit number was obtained. Find such a number.

b) In a three-digit number, the middle digit was erased and the resulting number was 6 times smaller than the original. Find such a three-digit number.

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Let $x, y, z$ be the digits of the desired number.

a) $100 x + 10 y + z = 7(10 y + z)$, from which $50 x = 3(10 y + z)$. Therefore, $x$ is divisible by 3. Since $3(10 y + z) < 300$, then $x = 3, 10 y + z = 50$.

b) $100 x + 10 y + z = 6(10 x + z)$, from which $8 x + 2 y = z$. The last equality is true only when $x = 1, y = 0, z = 8$.