Find the least value of \((4K^2 - 3)\) for which the six digits number 8768K6 must be divisible by 4.
1
A number is divisible by 4 when the number formed by its last two digits is divisible by 4. For 8768K6 that number is K6, i.e. 10K + 6.
Testing the digits: 06, 26, 46, 66, 86 are not divisible by 4, while 16, 36, 56, 76, 96 are. So K must be odd: K = 1, 3, 5, 7 or 9.
Since \(4K^2 - 3\) increases with K, its least value comes from the least K = 1: \(4(1)^2 - 3 = 4 - 3 = 1\).
Check: with K = 1 the number is 876816, whose last two digits 16 = 4 × 4, so 876816 is divisible by 4 (876816 = 4 × 219204); K = 0 gives 876806, which ends in 06 and is not divisible by 4.
Hence, 1.
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select the correct answer using the code given below:
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