Amrita needs to select a new PIN.
She decides it will be made up of four non-zero digits with the following properties:
i) The first two digits and the last two digits each make up a two-digit number which is a multiple of 11.
ii) The sum of all the digits is a multiple of 11.
How many different possibilities are there for Amrita’s PIN?
A. 1
B. 2
C. 4
D. 8
E. 16
D. 8
As the first two digits and the last two digits make up two two-digit numbers that are both multiples of 11, for example 33 and 55, the sum of the digits of the PIN is even. Note that the largest possible digit sum of a four-digit number is 36. Therefore, since the sum of the digits of the PIN is an even multiple of 11, that sum must be 22. So there are eight possibilities for Amrita’s PIN: 2299, 3388, 4477, 5566, 6655, 7744, 8833, 9922.
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