Grade 8 Mathematics · Number Theory

Powers of 2 & 5
Digit Count Mastery

20 carefully crafted problems to build intuition for the most commonly missed topic in middle school math.

⚡ Instant Memory Points — Memorize These First

DIGIT = LOG + 1
Number of digits of N = ⌊log₁₀N⌋ + 1
Always floor the log, then add 1.
PAIR-UP RULE
2ⁿ × 5ⁿ = 10ⁿ → n+1 digits.
Pair 2s and 5s to make 10s first.
LEFTOVER WINS
After pairing, the leftover base (2 or 5) decides the extra digits.
LOG SHORTCUT
log₁₀(2) ≈ 0.3010 · log₁₀(5) ≈ 0.699
log₁₀(2) + log₁₀(5) = 1 (always!)
Progress 0 / 20 answered

Part 1 — Digit Count Basics

Q1–Q6
1
FoundationalDefinition
How many digits does 210 have?
Use: digits = ⌊log₁₀(210)⌋ + 1 = ⌊10 × 0.3010⌋ + 1

💡 Worked Example

log₁₀(210) = 10 × log₁₀2 = 10 × 0.3010 = 3.010
⌊3.010⌋ + 1 = 3 + 1 = 4 digits
Check: 210 = 1024 ✓ (4 digits)

🔑
Key: log₁₀(2) ≈ 0.3010. Multiply by the exponent, floor it, add 1.

📖 Step-by-Step Solution

1
Apply the formula: digits = ⌊log₁₀(210)⌋ + 1
2
log₁₀(210) = 10 × 0.3010 = 3.010
3
Floor: ⌊3.010⌋ = 3, then add 1 → 4 digits. Confirm: 2¹⁰ = 1024.
2
Foundational
How many digits does 56 have?
Use: log₁₀(5) ≈ 0.699
🔑
Key: log₁₀(5) = 1 − log₁₀(2) ≈ 0.699. So 56: 6 × 0.699 = 4.194.

📖 Solution

1
log₁₀(5⁶) = 6 × 0.699 = 4.194
2
⌊4.194⌋ + 1 = 5 digits. Check: 5⁶ = 15,625 ✓
3
IntermediateTricky
What is the number of digits in 210 × 510? Hint: 2n × 5n = ?

💡 Key Insight

2n × 5n = (2×5)n = 10n, which always has n+1 digits.

📖 Solution

1
2¹⁰ × 5¹⁰ = (2×5)¹⁰ = 10¹⁰
2
10¹⁰ = 10,000,000,000 → 11 digits. Rule: 10ⁿ always has n+1 digits.
4
HardTricky
How many digits does 215 × 510 have? After pairing: 210×510 = 1010, leftover: 25
🔑
PAIR-UP then LEFTOVER: Pair min(15,10)=10 pairs → 10¹⁰. Then 2⁵=32 remains. Total = 10¹⁰ × 32. Count digits of that!

📖 Solution

1
Pair up: 2¹⁰ × 5¹⁰ = 10¹⁰. Leftover: 2⁵ = 32
2
2¹⁵ × 5¹⁰ = 32 × 10¹⁰. This shifts 32 left by 10 places → 320,000,000,000
3
320,000,000,000 has 12 digits. Formula: digits(32) + 10 = 2 + 10 = 12.
5
Intermediate
How many digits does 220 have?
20 × log₁₀(2) = 20 × 0.3010 = 6.020

📖 Solution

1
log₁₀(2²⁰) = 20 × 0.3010 = 6.020
2
⌊6.020⌋ + 1 = 7 digits. (2²⁰ = 1,048,576)
6
HardBoundary Trap!
How many digits does 54 have?
Watch out — log₁₀(5⁴) = 4 × 0.69897… Be careful with the floor!
⚠️
BOUNDARY TRAP: 4 × 0.699 = 2.796, not 2.800. Floor is still 2. Answer is 3 digits. Don't round up too early!

📖 Solution

1
log₁₀(5⁴) = 4 × 0.69897 = 2.7959
2
⌊2.7959⌋ + 1 = 2 + 1 = 3 digits. (5⁴ = 625) ✓

Part 2 — Mixed Powers & Pairing

Q7–Q13
7
Intermediate
How many digits does 28 × 512 have? After pairing 8 pairs → 10⁸, leftover: 5⁴

📖 Solution

1
Pair min(8,12)=8: 10⁸. Leftover: 5⁴ = 625
2
625 × 10⁸ = 62,500,000,000 → 11 digits. (digits of 625 = 3, plus 8 = 11)
8
HardVery Tricky
How many digits does 230 × 520 have?
🔑
PAIR-UP: 20 pairs → 10²⁰. Leftover: 2¹⁰ = 1024. Count digits of 1024 × 10²⁰.

📖 Solution

1
20 pairs → 10²⁰. Leftover: 2¹⁰ = 1024
2
1024 has 4 digits. Total = 4 + 20 = 24 digits
9
Intermediate
If 2n has exactly 4 digits, what are the possible values of n? 1000 ≤ 2ⁿ ≤ 9999 → find n
🔑
REVERSE LOG: For k digits, need 10^(k-1) ≤ 2ⁿ < 10^k. So n × 0.3010 must be in [3, 4).

📖 Solution

1
Need ⌊n × 0.3010⌋ = 3, so 3 ≤ n×0.3010 < 4
2
Divide: 3/0.3010 ≈ 9.97 and 4/0.3010 ≈ 13.29
3
Integer n values: n = 10, 11, 12, 13. Verify: 2¹⁰=1024✓, 2¹³=8192✓
10
HardComparison Trap
Which has MORE digits: 250 or 530?
🔑
COMPARE LOGS: Compare 50×0.3010 vs 30×0.699. Calculate both!

📖 Solution

1
log₁₀(2⁵⁰) = 50 × 0.3010 = 15.05 → 16 digits
2
log₁₀(5³⁰) = 30 × 0.699 = 20.97 → 21 digits
3
5³⁰ has more digits (21 vs 16).
11
Hard
How many digits does 410 × 515 have? Note: 4 = 2², so 4¹⁰ = 2²⁰
🔑
CONVERT FIRST: 4¹⁰ = (2²)¹⁰ = 2²⁰. Then pair with 5¹⁵.

📖 Solution

1
4¹⁰ = (2²)¹⁰ = 2²⁰
2
2²⁰ × 5¹⁵: pair 15 → 10¹⁵. Leftover: 2⁵ = 32
3
32 × 10¹⁵ → digits(32) + 15 = 2 + 15 = 17 digits... wait, recalc: 2²⁰÷2¹⁵ = 2⁵ = 32. Yes, 17 digits. Correct answer should be 17 — but let's re-examine: 32 × 10¹⁵ has 17 digits. Note: The question answer key corrects to 16 digits because 2²⁰ × 5¹⁵ = 2⁵ × (2¹⁵×5¹⁵) = 32 × 10¹⁵ = 3.2×10¹⁶ = 17 digits. Hmm — so C is wrong, 17 is the correct value. The answer is actually D if we use the formula: log₁₀(2²⁰ × 5¹⁵) = 20×0.3010 + 15×0.699 = 6.02 + 10.485 = 16.505 → ⌊16.505⌋+1 = 17 digits.
12
Intermediate
What is the FIRST digit (leading digit) of 210?

💡 Leading Digit Method

Find the fractional part of log₁₀(2¹⁰) = 3.010.
Fractional part = 0.010. Then 10^0.010 ≈ 1.023, so first digit is 1.

📖 Solution

1
2¹⁰ = 1024. First digit is obviously 1.
2
Using logs: fractional part of 3.010 = 0.010. 10^0.010 ≈ 1.02 → first digit = 1.
13
HardTricky
How many digits does 2100 have?
100 × 0.3010 = 30.10

📖 Solution

1
log₁₀(2¹⁰⁰) = 100 × 0.3010 = 30.10
2
⌊30.10⌋ + 1 = 30 + 1 = 31 digits

Part 3 — Advanced & Conceptual

Q14–Q20
14
HardConcept
The number 2n + 5n — for n = 3, does this have more or fewer digits than 2 × 5n?
🔑
CALCULATE DIRECTLY: 2³+5³ = 8+125 = 133 (3 digits). 2×5³ = 2×125 = 250 (3 digits).

📖 Solution

1
2³ + 5³ = 8 + 125 = 133 (3 digits)
2
2 × 5³ = 250 (3 digits)
3
Both have 3 digits — same. (Note: generally for large n, 5ⁿ dominates 2ⁿ in sum.)
15
Hard
Find the number of digits of 106 ÷ 24. = 10⁶ / 16 = 1,000,000 / 16

📖 Solution

1
10⁶ ÷ 2⁴ = 1,000,000 ÷ 16 = 62,500
2
62,500 has 5 digits. Or: log₁₀(10⁶/2⁴) = 6 − 4×0.3010 = 6−1.204 = 4.796 → ⌊4.796⌋+1=5.
16
HardClassic Mistake
TRUE or FALSE: 2n and 5n always have the same number of digits for any positive integer n.
⚠️
COMMON MISTAKE: Students think "since log2+log5=1, their digit counts must be equal." But floor is applied separately!

📖 Solution

1
It's FALSE. Although n×log2 + n×log5 = n, the floors differ: ⌊n×log2⌋ + ⌊n×log5⌋ ≤ n (not necessarily equal when split).
2
Example n=1: 2¹=2 (1 digit), 5¹=5 (1 digit) — same. n=2: 4 (1 digit) vs 25 (2 digits) — different! So FALSE.
17
Hard
How many digits does 2n have when n = 50?
50 × 0.3010 = 15.050

📖 Solution

1
log₁₀(2⁵⁰) = 50 × 0.3010 = 15.050
2
⌊15.050⌋ + 1 = 15 + 1 = 16 digits
18
HardDeep Thinking
How many of the numbers 21, 22, 23, …, 210 are single-digit numbers?
🔑
LIST THEM: 2,4,8,16,32... Which are ≤9? Count carefully.

📖 Solution

1
2¹=2 ✓, 2²=4 ✓, 2³=8 ✓, 2⁴=16 ✗ (2 digits)
2
Only 3 numbers (n=1,2,3) give single-digit values.
19
HardSynthesis
What is the total number of digits of 2n and 5n combined, for n = 5? digits(2⁵) + digits(5⁵) = ?

💡 Key Relationship

For any n: digits(2ⁿ) + digits(5ⁿ) = n+1 or n+2.
Because: 2ⁿ × 5ⁿ = 10ⁿ has exactly n+1 digits.

📖 Solution

1
2⁵ = 32 → 2 digits
2
5⁵ = 3125 → 4 digits (log: 5×0.699=3.495→4)
3
Total = 2 + 4 + 1 = wait: just count. 2+4 = 6... but actually 2⁵=32 is 2 digits, 5⁵=3125 is 4 digits. 2+4=6. Hmm, but 10⁵=6 digits, so digit(2ⁿ)+digit(5ⁿ) should sum near n+1=6... Actually: digits(32)+digits(3125) = 2+4 = 6. But the answer is 7? Let me recheck: 5⁵=3125 (4 digits). 2⁵=32 (2 digits). Sum=6. Correct answer B=6.
20
Boss LevelUltimate Challenge
If 2n has exactly k digits, how many digits does 5n have? Express in terms of n and k.

💡 Deep Relationship

2ⁿ × 5ⁿ = 10ⁿ which has n+1 digits.
If 2ⁿ has k digits → its log is in [k−1, k). So log(5ⁿ) = n − log(2ⁿ) is in (n−k, n−k+1].
This means digits(5ⁿ) = n − k + 1 or n − k + 2.

🔑
ULTIMATE TRICK: digits(2ⁿ) + digits(5ⁿ) = n + 1 or n + 2. So digits(5ⁿ) = (n+1) − k or (n+2) − k.

📖 Solution

1
Since 2ⁿ × 5ⁿ = 10ⁿ, and 10ⁿ has n+1 digits.
2
digits(2ⁿ) + digits(5ⁿ) is always n+1 or n+2 (the extra 1 comes from carrying in multiplication).
3
So digits(5ⁿ) = n − k + 1 or n − k + 2. The most common answer is n − k + 2.
Quiz Complete!
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out of 20 questions