Number System and Digits

The unit digit of (2^31+3)^25 + (7^29-3)^23 is 5. Breaking it down: (2^31+3) has unit digit (8+3)=11→1, so 1^25=1. (7^29-3) has unit digit (7-3)=4, and 4^23 has unit digit 4 (since 4^odd=4). Adding: 1+4=5. This tests understanding of unit digit patterns in powers.

If A is even, A^k ends in 6 if k is a multiple of 4 (except for A ending in 0). If A ends in 0, A^k ends in 0. Since 4n-4 is a multiple of 4 for n>1, the unit digit must be 0 or 6.

Counting letters in English digit names: zero(4), one(3), two(3), three(5), four(4), five(4), six(3), seven(5), eight(5), nine(4). Only 'four' has exactly 4 letters, equal to its numerical value.

The local number has 7 digits where the first digit cannot be 0 (9 choices: 1-9), and the remaining 6 digits can be 0-9 (10 choices each). Total combinations = 9 × 10^6 = 9,000,000. Option B (8,000,000) incorrectly assumes 8 choices for the first digit. Options C and D add arbitrary numbers not justified by the counting principle.

The number of combinations is calculated by multiplying the possibilities for each slot. There are $26^3$ ways for the letters and $10^4$ ways for the digits. $17,576 \times 10,000 = 175,760,000$.

A remainder $< 10$ when divided by 100 means the tens digit 'a' must be 0. The number is $2130b$. The sum of digits is $2+1+3+0+b = 6+b$. If the sum is 13, then $6+b=13$, so $b=7$.

The product contains 10 as a factor, which guarantees the last digit is 0. When you multiply any integer by 10, the result ends in 0. This is a pattern recognition question - you don't need to calculate the full product.

Pages 1-9: 9 digits. Pages 10-99: 90 * 2 = 180 digits. Pages 100-366: 267 * 3 = 801 digits. Total: 9 + 180 + 801 = 990. The other options like 366 or 900 are incorrect counts of the digit distribution.

For 84: the second digit (4) is smaller than the first (8). The sum of digits is 8+4=12. 84 divided by 12 equals 7. This satisfies all conditions. Other options like 62 (62/8) or 73 (73/10) do not result in a quotient of 7.

Standard Visa card numbers always contain 16 digits. This is part of the ISO/IEC 7812 standard for payment cards. MasterCard also uses 16 digits, while American Express uses 15 digits.

For 12: reversed is 21 (21-12=9). Sum of digits (3) is a factor of 12 and 21. Product of digits (2) is a factor of 12 but not 21. All conditions are met.

Let the tens digit be t and units digit be u. Given u = t + 1 and 10t + u = 4(t + u). Substituting: 10t + t + 1 = 4(2t + 1), so 11t + 1 = 8t + 4, giving 3t = 3 and t = 1. Therefore u = 2, and the number is 12. Option D is correct.

Hiroyuki Goto recited pi to 42,000 digits in 1995, setting a then-world record. This feat took over 9 hours to complete and demonstrated extraordinary memorization ability. The other options (12,000; 24,000; 56,000) are incorrect - 42,000 was his specific verified achievement.

Testing each missing digit in 3170?4847 with the Dutch bank account rule (sum of digits × weights must be divisible by 11): For ?=3: 3×9 + 1×8 + 7×7 + 0×6 + 3×5 + 4×4 + 8×3 + 4×2 + 7×1 = 27+8+49+0+15+16+24+8+7 = 154. Since 154 ÷ 11 = 14 (exactly divisible), the missing digit is 3.

Let the number be 10a + b (two-digit number). Condition: 10a + b = 5(a + b), so 10a + b = 5a + 5b, giving 5a = 4b. Testing integer values: a=4, b=5 gives 45. Checking: 4+5=9, 9×5=45. Option C is correct.