Puzzle for April 15, 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.
* AB is a 2-digit number (not A×B).
Scratchpad
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Hint #1
Add B to both sides of eq.4: A – B + B = C + E + B which becomes A = C + E + B In eq.2, replace A with C + E + B: C = C + E + B + E Subtract C from each side of the equation above: C – C = C + E + B + E – C 0 = B + 2×E Since B and E are one-digit non-negative integers, the above equation makes B = 0 and E = 0
Hint #2
In eq.4, replace both B and E with 0: A – 0 = C + 0 which means A = C
Hint #3
eq.5 may be written as: C + D = 10×A + B Substitute A for C, and 0 for B in the above equation: A + D = 10×A + 0 Subtract A from each side: A + D – A = 10×A + 0 – A which makes D = 9×A
Hint #4
Substitute 9×A for D, and A for C in eq.3: 9×A – F = A + F Add F to both sides, and subtract A from each side of the above equation: 9×A – F + F – A = A + F + F – A which simplifies to 8×A = 2×F Divide each side by 2: 8×A ÷ 2 = 2×F ÷ 2 which makes 4×A = F
Solution
Substitute 0 for B and E, A for C, 9×A for D, and 4×A for F in eq.1: A + 0 + A + 9×A + 0 + 4×A = 15 which simplifies to 15×A = 15 Divide both sides by 15: 15×A ÷ 15 = 15 ÷ 15 which means A = 1 making C = A = 1 D = 9×A = 9 × 1 = 9 F = 4×A = 4 × 1 = 4 and ABCDEF = 101904