Question

N = abcd is a four digit number and M = xyz is a three digit number. \(N \times M\) is:

• A. a five digit number when a = x = 1
• B. a six digit number when a = 3 and x = 3
• C. a seven digit number when a = 6 and x = 2
• D. an eight digit number when a = b = c = d = x = y = z = 9

Hint:

Fix the leading digits to get the smallest and largest possible product, then count digits at both ends of that range. Only option (3) keeps the same digit count throughout.

Answer 1

The Correct Option is C

There is a quick way to count digits without multiplying everything out: the number of digits in a positive whole number M is $\lfloor \log_{10} M \rfloor + 1$. We can use the boundary values of N and M in each option to see whether the digit count of the product stays fixed across the whole range.

1. Option 1 (a = x = 1): N is between $1000$ and $1999$, M is between $100$ and $199$. The smallest product is $1000 \times 100 = 100000$, already a 6 digit number. Since the smallest case already gives 6 digits, the product is never only 5 digits. Wrong.
2. Option 2 (a = x = 3): N is between $3000$ and $3999$, M is between $300$ and $399$. Smallest product $3000 \times 300 = 900000$ (6 digits), largest product $3999 \times 399 = 1595601$ (7 digits, since it passes 1,000,000). The digit count changes within the range, so it is not always 6 digits. Wrong.
3. Option 3 (a = 6, x = 2): N is between $6000$ and $6999$, M is between $200$ and $299$. Smallest product: $6000 \times 200 = 1200000$. Largest product: $6999 \times 299 = 2092701$. Both the smallest and the largest value lie strictly between 1,000,000 and 9,999,999, the range of all 7 digit numbers, so the product is always 7 digits here. Correct.
4. Option 4 (all nines): N = $9999$, M = $999$, so the product is $9999 \times 999 = 9989001$, which is only 7 digits, not 8. Wrong.

Since the product's smallest and largest values in option (3) both fall strictly inside the 7 digit range, and the product increases smoothly as N and M increase, every value of $N \times M$ in that case has exactly 7 digits.

Let's summarize:

• Digit count of a number can be checked from its size relative to powers of 10, without multiplying everything out.
• Options 1, 2 and 4 all fail because their product's digit count is not fixed across the whole range, or does not match the number of digits claimed.
• Option 3 is the only case where both the smallest and largest product share the same digit count, 7.

So $N \times M$ is a seven digit number when a = 6 and x = 2, which is option (3).