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Number Theory : Remainders, Divisors

Remainders of division of two different numbers and their sum by the same divisor

Question

Let n be the number of different 5 digit numbers, divisible by 4 with the digits 1, 2, 3, 4, 5 and 6, no digit being repeated in the numbers. What is the value of n?

(1) 144
(2) 168
(3) 192
(4) None of these

Correct Choice is (3) and correct answer is 192

Explanatory Answer

Test of divisibility by 4 is that the last two digits should be divisible by 4.

Case 1 : When the last 2 digits are 12, i.e., _ _ _ 12 = 4 * 3 * 2 = 24 numbers

Case 2 : When the last 2 digits are 16, there are 24 numbers

Case 3 : When the last 2 digits are 24 there are 24 numbers

Case 4 : When the last 2 digits are 32 there are 24numbers

Case 5 : When last 2 digits are 36 there are 24 numbers

Case 6 : When last 2 digits are 52 there are 24 numbers

Case 7 : When last 2 digits are 56 there are 24 numbers

Case 8 : When last 2 digits are 64 there are 24 numbers

Total = 8 * 24 = 192

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