1. **Stating the problem:** We want to find a way to get the numbers 9, 17, 36, 47, and 64 starting from 87 by repeatedly adding either 1 or 2. 2. **Understanding the problem:** Starting at 87, we want to reach each target number by adding 1 or 2 multiple times. Since 9, 17, 36, 47, and 64 are less than 87, this suggests we might be subtracting 1 or 2 instead of adding to get down to those numbers. 3. **Reformulating:** To get from 87 down to each target number by adding negative 1 or negative 2 (equivalent to subtracting 1 or 2), we check if the difference between 87 and each target can be expressed as a sum of 1s and 2s. 4. **Calculate differences:** $$ 87 - 9 = 78 $$ $$ 87 - 17 = 70 $$ $$ 87 - 36 = 51 $$ $$ 87 - 47 = 40 $$ $$ 87 - 64 = 23 $$ 5. **Check if each difference can be formed by sums of 1s and 2s:** Since any positive integer can be expressed as a sum of 1s and 2s, it is possible to reach each target number from 87 by subtracting 1 or 2 repeatedly. 6. **Example for 9:** $$ 78 = 2 \times 39 $$ So, subtract 2 thirty-nine times. 7. **Example for 17:** $$ 70 = 2 \times 35 $$ Subtract 2 thirty-five times. 8. **Example for 36:** $$ 51 = 2 \times 25 + 1 $$ Subtract 2 twenty-five times and then subtract 1 once. 9. **Example for 47:** $$ 40 = 2 \times 20 $$ Subtract 2 twenty times. 10. **Example for 64:** $$ 23 = 2 \times 11 + 1 $$ Subtract 2 eleven times and then subtract 1 once. **Final answer:** You can get from 87 to each of the numbers 9, 17, 36, 47, and 64 by subtracting 1 or 2 repeatedly, since the differences are all positive integers that can be expressed as sums of 1s and 2s.