Puzzle for June 17, 2019 ( )
Scratchpad
Find the 6-digit number ABCDEF by solving the following equations:
A, B, C, D, E, and F each represent a one-digit non-negative integer.
Scratchpad
Help Area
Hint #1
Add B to both sides of eq.5: C - B + B = D + E + B which becomes C = D + E + B which may be written as C = D + B + E In the above equation, replace B + E with C (from eq.3): C = D + C Subtract C from both sides: C - C = D + C - C which makes 0 = D
Hint #2
Add B to each side of eq.6: E - B + B = B + D + B which becomes E = 2×B + D Replace D with 0: E = 2×B + 0 which means E = 2×B
Hint #3
In eq.3, substitute 2×B for E: B + 2×B = C which makes 3×B = C
Hint #4
Substitute 0 for D, 3×B for C, and 2×B for E in eq.2: A + 0 = 3×B + 2×B which makes A = 5×B
Hint #5
Substitute 5×B for A, and 2×B for E in eq.4: F - B = 5×B + 2×B which becomes F - B = 7×B Add B to both sides: F - B + B = 7×B + B which makes F = 8×B
Solution
Substitute 5×B for A, 3×B for C, 0 for D, 2×B for E, and 8×B for F in eq.1: 5×B + B + 3×B + 0 + 2×B + 8×B = 19 which simplifies to 19×B = 19 Divide both sides by 19: 19×B ÷ 19 = 19 ÷ 19 which means B = 1 making A = 5×B = 5 × 1 = 5 C = 3×B = 3 × 1 = 3 E = 2×B = 2 × 1 = 2 F = 8×B = 8 × 1 = 8 and ABCDEF = 513028