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Home » Thus, number with **at least one** "33" is $ 64 - 21 = 43 $ - High Altitude Science

Thus, number with at least one "33" is $ 64 - 21 = 43 $ - High Altitude Science

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 So we solve \( 9x^2 + 16y^2 = 144 \) in integers with \( |x| \leq 4 \), \( |y| \leq 3 \).

📌 We try all integer \( x \) from \( -4 \) to \( 4 \), compute \( 9x^2 \), then \( 16y^2 = 144 - 9x^2 \), so \( y^2 = \frac{144 - 9x^2}{16} \), must be a perfect square and integer.

📌 \( x = 0 \): \( y^2 = \frac{144}{16} = 9 \) → \( y = \pm 3 \) → valid

📌 \( x = \pm 1 \): \( 9(1) = 9 \), \( 144 - 9 = 135 \), \( y^2 = 135/16 \) → not integer

📌 \( x = \pm 2 \): \( 9(4) = 36 \), \( 144 - 36 = 108 \), \( y^2 = 108/16 = 27/4 \) → not integer

📌 \( x = \pm 3 \): \( 9(9) = 81 \), \( 144 - 81 = 63 \), \( y^2 =

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