what two numbers make a one-digit number when you multiply them but make a two-digit number when you add them together?

Let's analyze the problem step-by-step: You want two numbers such that:

1. When you multiply them, the result is a one-digit number (i.e., between 0 and 9).
2. When you add them, the result is a two-digit number (i.e., between 10 and 99).

Step 1: Understand the constraints

• Multiplication result: 1 to 9 (assuming positive integers)
• Addition result: 10 to 99

Step 2: Try to find such pairs

Let's denote the two numbers as xxx and yyy.

• x×y≤9x\times y\leq 9x×y≤9
• x+y≥10x+y\geq 10x+y≥10

Step 3: Explore possibilities

Since the product is small (≤9), both numbers must be relatively small or one must be 1. Try some pairs:

• (1, 9):

  • Product: 1×9=91\times 9=91×9=9 (one-digit ✓)
  • Sum: 1+9=101+9=101+9=10 (two-digit ✓)
    This pair works!

• (2, 5):

  • Product: 2×5=102\times 5=102×5=10 (two-digit ✗)
    No.

• (3, 4):

  • Product: 3×4=123\times 4=123×4=12 (two-digit ✗)
    No.

• (1, 8):

  • Product: 1×8=81\times 8=81×8=8 (one-digit ✓)
  • Sum: 1+8=91+8=91+8=9 (one-digit ✗)
    No.

• (1, 10):

  • Product: 1×10=101\times 10=101×10=10 (two-digit ✗)
    No.

Conclusion:

The pair (1, 9) satisfies the condition perfectly:

• Multiply: 1×9=91\times 9=91×9=9 (one-digit number)
• Add: 1+9=101+9=101+9=10 (two-digit number)

If you want other pairs, you can try (1, any number between 9 and 98), but the product will be larger than 9 once the second number is greater than 9, so only (1, 9) works for positive integers. Answer:
The two numbers are 1 and 9.