Puzzle for May 27, 2019 ( )
Scratchpad
Find the 4-digit number ABCD by solving the following equations:
A, B, C, and D each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
In eq.4, replace A with B + C + D (from eq.2): D - B = B + C + D + B - C - D which becomes D - B = 2×B Add B to both sides of the above equation: D - B + B = 2×B + B which makes D = 3×B
Hint #2
In eq.3, replace D with 3×B: B = C - 3×B Add 3×B to both sides of the above equation: B + 3×B = C - 3×B + 3×B which means 4×B = C
Hint #3
Substitute 4×B for C, and 3×B for D in eq.2: A = B + 4×B + 3×B which makes A = 8×B
Solution
Substitute 8×B for A, 4×B for C, and 3×B for D in eq.1: 8×B + B + 4×B + 3×B = 16 which simplifies to 16×B = 16 Divide each side by 16: 16×B ÷ 16 = 16 ÷ 16 which means B = 1 making A = 8×B = 8 × 1 = 8 C = 4×B = 4 × 1 = 4 D = 3×B = 3 × 1 = 3 and ABCD = 8143