Find the lowest positive value of (c – b) such that the 7-digit number 1738b9c is divisible by 12.

Given:

A 7-digit number 1738b9c

Formula used:

A number is divisible by 12 if and only if it is divisible by both 3 and 4.

Calculations:

Divisibility by 3: Sum of the digits must be divisible by 3.

1 + 7 + 3 + 8 + b + 9 + c = 28 + b + c

⇒ 28 + b + c must be divisible by 3

Divisibility by 4: Last two digits must be divisible by 4.

Last two digits = 9c

⇒ 9c must be divisible by 4

Possible values for c such that 9c is divisible by 4:

c = 2 or c = 6

Case 1: c = 2

28 + b + 2 = 30 + b

⇒ 30 + b must be divisible by 3

⇒ Possible values for b = 0, 3, 6, 9

c - b = 2 - 0 = 2

c - b = 2 - 3 = -1

c - b = 2 - 6 = -4

c - b = 2 - 9 = -7

Case 2: c = 6

28 + b + 6 = 34 + b

⇒ 34 + b must be divisible by 3

⇒ Possible values for b = 2, 5, 8

c - b = 6 - 2 = 4

c - b = 6 - 5 = 1

c - b = 6 - 8 = -2

∴ The lowest positive value of (c - b) is 1 when c = 6 and b = 5.