Problem Solving

This is a COMBINATORICS problem.

If we're looking for a prime tens and units digit, we're counting single-digit prime numbers: 2, 3, 5, and 7.

Now we need to count the possibilities for each digit. Since we have the constraint that we're looking for all #s 0-1570, we have to be careful, since we can't always set constraints unilaterally on a given digit. We can set a unilateral constraint on the thousands digit (must be a 0 or 1), but not on the hundreds digit. E.g. the digit 9 is a possibility for the hundreds digit... but only if the thousands digit is 0, or we'd have 1900+.

So it's easiest to count in 2 parts:

1. The number of 4-digit numbers less than 1570 with prime tens and units digits:
thousands digit: 1 possibility (thousands digit must be 1)
hundreds digit: 6 possibilities (0, 1, 2, 3, 4, or 5)
tens digit: 4 possibilities (2, 3, 5, 7)
units digit: 4 possibilities (2, 3, 5, 7)

Now we multiply the # of possibilities for each digit:
$$1\cdot6\cdot4\cdot4=96$$
But... this number includes a few possibilities that don't work because they're greater than 1570: 1577, 1575, 1573, 1572. We simply need to exclude these 4 possibilities by subtracting them from the total:
96 - 4 = 92.

(Note: we wouldn't want to exclude 7 unilaterally as a possibility for the tens digit, since 1477 or 1372 would have worked, etc).

2. Now, count the number of 2- and 3-digit numbers with prime tens and units digits:
hundreds digit: 10 possibilities (any digit 0-9, giving us a 2- or 3-digit number)
tens digit: 4 possibilities (2, 3, 5, 7)
units digit: 4 possibilities (2, 3, 5, 7)
$$10\cdot4\cdot4=160$$

Since all of these possibilities are between 0 and 1570, we don't need to exclude any.

Now simply add the two parts together:
92 + 160 = 252

The answer is B.

BTGmoderatorLU wrote:Source: Princeton Review

How many integers between 0 and 1570 have a prime tens digit and a prime units digit?

A. 295
B. 252
C. 236
D. 96
E. 76

The OA is B

Since 4 digits are prime digits (2, 3, 5, 7), if the number has 2 digits, then we have 4 x 4 = 16 such numbers. If the number has 3 digits, then we have 9 x 4 x 4 = 144 such numbers. If the number has 4 digits and it's less than 1500, then we have 1 x 5 x 4 x 4 = 80 such numbers. Finally, if the number has 4 digits and it's between 1500 and 1570 (inclusive), then we have 1 x 1 x 3 x 4 = 12 such numbers. Therefore, we have a total of 16 + 144 + 80 + 12 = 252 numbers between 0 and 1570 that have a prime tens digit and a prime units digit.

Answer: B