We will say that a positive integer n is subject to an interesting change if it is multiplied by 2 and the result is increased by 4, a special change if it is multiplied by 3 and the result is increased by 9 and an awesome change if it is multiplied by 4 and the result is increased by 16.
a) Show that there exists a positive integer which after three changes, the first – interesting, the second – special and the third – awesome, becomes 2020.
b) Find all positive integers with the property that after two changes of different types, selected among the three above, becomes 2014.
Solution
a) Before the last change the number must be (2020−16):4=501; the previous number must be (501−9):3=164 and the required number is (164−4):2=80.
b) An interesting change produces a multiple of 2, a special change gives a multiple of 3 and an awesome change yields a multiple of 4. Since 2014 is neither a multiple of 3, nor a multiple of 3, the last change must be an interesting one. So, the second number must be (2014−4):2=1005. Since this number is odd, the first change must be special, and the initial number is (1005−9):3=332.
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