Calling multiples of 4

Number Theory Level 2

Calvin wishes to make a phone call to a friend. He knows that the first 3 digits of the 7-digit phone number are 765, and the other 4 digits consists of the digits 4, 3, 2 and 1 in some order. He recalls that the phone number is a multiple of 4. How many possible telephone numbers are there, which satisfy these conditions?

The answer is 6.

2 solutions

Arron Kau Staff
May 13, 2014

A number is a multiple of 4 if and only if the last 2 digits are a multiple of 4. From the conditions given, the last 2 digits must be either $12, 32$ or $24$ . In each of these cases, the other 2 digits can be placed in any order, giving $2 \times 1$ possibilities. Hence, in total, there are $3 \times 2 = 6$ possibilities.

We can explicitly list these out as $3412$ , $4312$ , $1432$ , $4132$ , $1324$ , $3124$ .

Shreya R
Oct 24, 2014

The divisibility rule for 4 is that the number formed by the last two digits must be divisible by 4. So in the last two places, either 32, 12 or 24 has to be present and the preceding digits can be arranged i any way. This gives us $2*3$ , that is $6$ solutions.

Calling multiples of 4

The answer is 6.

2 solutions

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