Number System — HCF / LCM / Divisibility / Remainders / Factors

Section A · Futures First International — Market Trading Intern OA & Interview Guide

📌 What Has Been Asked (OA, Buzzer Round & Interviews)

Futures First assesses quantitative reflexes under extreme time constraints (30–45 seconds per question in OA, 3–5 seconds in buzzer/group rounds, and verbal explanations in interviews). Here are the exact recurring question patterns:

1. Online Assessment (OA) Recurring Archetypes

Topic	Real OA Problem Style	Frequency
Compound Divisibility	"If `5432y17x` is divisible by 72, find the values of $(x + y)$" (or divisible by 88, 99)	Very High
HCF Difference Model	"Find the greatest number that divides 112, 168, and 280 leaving the same remainder in each case"	High
LCM Bells / Racing	"Three bells toll at intervals of 9, 12, 15 minutes. If they toll together now, when will they toll together next?"	High
LCM Remainder Models	"Find the smallest number which when divided by 12, 15, 20 leaves remainder 4 in each case" OR "leaves remainders 8, 11, 16"	Very High
Modular Powers	"What is the remainder when $17^{200}$ is divided by 18?" or "Remainder when $2^{100}$ is divided by 13?"	Very High
Negative Remainder / Binomial	"Find remainder when $(35^{37} + 1)$ is divided by 36"	High
Units Digit with Borrowing	"Find the unit digit of $(7^{95} - 3^{58})$"	High
Trailing Zeros	"Find the number of trailing zeros in $125!$" or "in $25! \times 35! \times 45!$"	High
Power in Factorial	"What is the highest power of 6 (or 12) that divides $50!$?"	High
Factor Counts	"Find the number of odd factors and even factors of 720"	Medium

2. Group Round / "Fastest-Finger" Buzzer Questions

"Is 91 prime?" $\rightarrow$ Instant answer: No, $7 \times 13 = 91$ (a notorious trading test trap!).
"Is 143 prime?" $\rightarrow$ No, $11 \times 13 = 143$.
"Is 221 prime?" $\rightarrow$ No, $13 \times 17 = 221$.
"Smallest 3-digit prime number?" $\rightarrow$ 101.
"A number has exactly 3 factors. What kind of number is it?" $\rightarrow$ The square of a prime number ($p^2$ has factors $1, p, p^2$).
"What is the unit digit of $2^{43}$?" $\rightarrow$ $43 \pmod 4 = 3 \implies 2^3 = \mathbf{8}$.
"Is 7,392 divisible by 8?" $\rightarrow$ Check 392: $392/8 = 49 \implies \mathbf{Yes}$.

3. Trader Stress Interview Questions

"Why do we count only 5s and not 2s when finding trailing zeros in $n!$?"
- Answer: Every trailing zero is formed by $2 \times 5$. In any factorial, multiples of 2 appear far more frequently than multiples of 5, making 5 the limiting factor.
"How do you calculate $7^{100} \pmod{11}$ verbally in under 5 seconds?"
- Answer: By Fermat’s Little Theorem, $7^{10} \equiv 1 \pmod{11}$. Since $100$ is a multiple of $10$, $(7^{10})^{10} \equiv 1^{10} = \mathbf{1}$.
"If a stock ticks every 6 seconds, another every 8 seconds, and a third every 14 seconds, how many times will they tick simultaneously in a 14-minute trading window?"
- Answer: $\text{LCM}(6, 8, 14) = 168$ seconds. $14 \text{ min} = 840 \text{ sec}$. $840 / 168 = \mathbf{5 \text{ times}}$ (plus initial tick if inclusive).

🧠 Core Concepts & How to Solve Step-by-Step

1. HCF & LCM — The 4 Core Models

Fundamental Property:

$$\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b$$ (Applies strictly to two numbers; for 3+ numbers, use prime factorization!)

Fractions:

$$\text{HCF}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{HCF}(a, c)}{\text{LCM}(b, d)}, \quad \text{LCM}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{LCM}(a, c)}{\text{HCF}(b, d)}$$

🏆 Model A: Least Number Leaving Same Remainder $R$

Problem statement: "Smallest number which when divided by $x, y, z$ leaves remainder $R$ in each case."
Formula: $$\mathbf{N = \text{LCM}(x, y, z) \times k + R}$$
For smallest positive integer, set $k = 1$.

🏆 Model B: Least Number Leaving Different Remainders with Constant Difference

Problem statement: "Smallest number which when divided by $x, y, z$ leaves remainders $r_1, r_2, r_3$ respectively," where: $$(x - r_1) = (y - r_2) = (z - r_3) = K \quad (\text{constant difference})$$
Formula: $$\mathbf{N = \text{LCM}(x, y, z) \times k - K}$$

🏆 Model C: Greatest Number Leaving the Same Remainder (Unknown Remainder)

Problem statement: "Greatest number that divides $x, y, z$ leaving the same remainder in each case."
Method (No equations needed!): $$\mathbf{HCF(|x - y|, |y - z|, |z - x|)}$$
The remainder $R$ is found simply by dividing any of the original numbers by this HCF.

🏆 Model D: Greatest Number Leaving Specific Remainders

Problem statement: "Greatest number that divides $x, y, z$ leaving remainders $r_1, r_2, r_3$."
Formula: $$\mathbf{HCF(x - r_1, y - r_2, z - r_3)}$$

2. Divisibility Rules & Compound Divisibility (72, 88, 99)

Rapid Divisibility Table:

Divisor	Rapid Test	Example / Fast Check
2	Last digit even	—
3	Sum of digits divisible by 3	Discard 3s, 6s, 9s as you add digits!
4	Last 2 digits divisible by 4	$7,316 \implies 16$ is div by 4 ✓
5	Last digit 0 or 5	—
6	Even AND sum of digits div by 3	—
7	Double last digit, subtract from remaining	$343 \implies 34 - 2(3) = 28$ (div by 7 ✓)
8	Last 3 digits divisible by 8	For $x$, if hundreds digit is even, test last 2 digits. If odd, test last 2 digits + 4!
9	Sum of digits divisible by 9	"Cast out 9s" (ignore pairs summing to 9)
11	$(\text{Sum of odd positions}) - (\text{Sum of even positions}) = 0 \text{ or multiple of } 11$	Alternate $+ - + -$ from left to right
13	Add 4 times last digit to remaining	$143 \implies 14 + 4(3) = 26$ (div by 13 ✓)

⚡ Compound Divisibility Trick (The OA Favorite):

When a question asks for divisibility by a composite number $N$, break $N$ into co-prime factors:

Divisible by 72: Check 8 (last 3 digits) AND 9 (sum of digits).
- Method: Always solve for the last digit using the rule of 8 first! Once that digit is fixed, sum all digits to solve for the other digit using the rule of 9.
Divisible by 88: Check 8 AND 11.
Divisible by 99: Check 9 AND 11 (or sum 2-digit blocks from the right: $12375 \implies 75 + 23 + 1 = 99$ ✓).

3. Remainders & Modular Arithmetic

Concept 1: Negative Remainders (The Speed King)

When a number is 1 less than the divisor, treat the remainder as $-1$: $$\frac{17}{18} \implies \text{Remainder } -1 \equiv 18 - 1 = 17$$ $$\frac{17^{200}}{18} \equiv (-1)^{200} = \mathbf{1}$$ $$\frac{17^{201}}{18} \equiv (-1)^{201} = -1 \equiv 18 - 1 = \mathbf{17}$$

Concept 2: Binomial Expansions for Remainder

$$\frac{(ax + 1)^n}{a} \implies \text{Remainder } = 1^n = \mathbf{1}$$ $$\frac{(ax - 1)^n}{a} \implies \text{Remainder } = (-1)^n = \begin{cases} 1 & \text{if } n \text{ is even} \ a - 1 & \text{if } n \text{ is odd} \end{cases}$$

Concept 3: Fermat’s Little Theorem

If $p$ is a prime number and $\gcd(a, p) = 1$: $$a^{p - 1} \equiv 1 \pmod p$$

How to use: Reduce the exponent $n$ modulo $(p - 1)$!
Example: Find remainder when $2^{100}$ is divided by 13.
- Divisor $p = 13$ (prime), so $p - 1 = 12$.
- Exponent $100 = 12 \times 8 + 4$.
- Remainder is simply $2^4 \pmod{13} = 16 \pmod{13} = \mathbf{3}$. (Solved in 5 seconds!)

Concept 4: Wilson's Theorem

If $p$ is a prime number: $$(p - 1)! \equiv -1 \equiv (p - 1) \pmod p$$ $$(p - 2)! \equiv 1 \pmod p$$

Example: Remainder when $28!$ is divided by 29 $\implies \mathbf{28}$ (or $-1$).
Example: Remainder when $27!$ is divided by 29 $\implies \mathbf{1}$.

4. Units Digit Cyclicity

The unit digit of $a^b$ depends only on the unit digit of $a$ and has a repeating pattern of length at most 4:

Last Digit of Base	Cycle Pattern	Cycle Length
0, 1, 5, 6	Always same: $0 \to 0$, $1 \to 1$, $5 \to 5$, $6 \to 6$	1
4	$4^1 = 4$, $4^2 = 6$ (Odd $\to$ 4, Even $\to$ 6)	2
9	$9^1 = 9$, $9^2 = 1$ (Odd $\to$ 9, Even $\to$ 1)	2
2	$2, 4, 8, 6$	4
3	$3, 9, 7, 1$	4
7	$7, 9, 3, 1$	4
8	$8, 4, 2, 6$	4

⚡ Step-by-Step Method:

Extract the last digit of the base.
Divide the power by 4 and find remainder $r = \text{power} \pmod 4$.
If $r = 1, 2, 3$, the unit digit is $(\text{base})^r$.
If $r = 0$, treat it as $(\text{base})^4$.
⚠️ Borrowing Trap: If calculating $(7^{95} - 3^{58})$:
- $7^{95} \implies 95 \pmod 4 = 3 \implies 7^3 = 34\mathbf{3}$ (unit digit 3).
- $3^{58} \implies 58 \pmod 4 = 2 \implies 3^2 = \mathbf{9}$ (unit digit 9).
- Result: $3 - 9 = -6 \implies$ You must borrow 10: $13 - 9 = \mathbf{4}$! (Never answer $-6$ or $6$!).

5. Factor Theory (Counts, Odd, Even, Sum)

Express $N$ in standard prime-factorized form: $$N = p_1^a \cdot p_2^b \cdot p_3^c$$ (where $p_1, p_2, p_3$ are distinct primes)

Metric	Formula	Mental Note
Total Number of Factors	$(a+1)(b+1)(c+1)$	Add 1 to each exponent and multiply
Number of Odd Factors	Exclude $2^a$, multiply remaining $(b+1)(c+1)$	Zero powers of 2 allowed
Number of Even Factors	$\text{Total Factors} - \text{Odd Factors} = a(b+1)(c+1)$	Must have at least one power of 2 ($2^1, \dots, 2^a$)
Sum of Factors	$\frac{p_1^{a+1}-1}{p_1-1} \times \frac{p_2^{b+1}-1}{p_2-1} \times \dots$	Formula for sum of geometric series
Product of Factors	$N^{(\text{Total Factors} / 2)}$	Pairs of factors multiplying to $N$
Number of Prime Factors	$(a + b + c)$	Count with multiplicities
Distinct Prime Factors	Number of unique bases (here, 3)	Just count $p_1, p_2, p_3$

6. Trailing Zeros & Legendre’s Formula

The number of trailing zeros in $n!$ equals the highest power of 5 dividing $n!$: $$E_5(n!) = \left\lfloor \frac{n}{5} \right\rfloor + \left\lfloor \frac{n}{5^2} \right\rfloor + \left\lfloor \frac{n}{5^3} \right\rfloor + \dots$$

⚡ Fast Mental Calculation:

Successively divide by 5 and add the quotients:

Trailing zeros in $100!$: $$100/5 = 20$$ $$20/5 = 4$$ $$20 + 4 = \mathbf{24}$$
Trailing zeros in $125!$: $$125/5 = 25$$ $$25/5 = 5$$ $$5/5 = 1$$ $$25 + 5 + 1 = \mathbf{31}$$

Composite Base Extension: Highest Power of 6, 12, etc. in $n!$:

To find highest power of 6 in $n!$: $6 = 2 \times 3$. The limiting prime is 3, so calculate $E_3(n!)$.
To find highest power of 12 in $n!$: $12 = 2^2 \times 3$. Calculate $E_2(n!)$ and $E_3(n!)$. Power of 12 is $\min(\lfloor E_2/2 \rfloor, E_3)$.

⚡ Speed Tricks & Flowchart for Futures First

                      [ Number System Question ]
                                   |
    -----------------------------------------------------------------
    |                       |                       |               |
[Divisibility]         [Remainder]              [HCF/LCM]       [Factorials]
    |                       |                       |               |
Compound (72, 88)?    Base near divisor?     Bells/Laps?      Trailing zeros?
Check 8 on last 3     Use (-1) or (+1)       Take LCM!        Divide by 5
digits first!         Negative remainder!           |         repeatedly.
    |                       |                Same remainder?        |
Digit sum for 9.      Prime divisor?         HCF of pairwise  Power of comp?
                      Fermat: a^(p-1) = 1    differences!     Limit by largest
                                                              prime factor!

🎯 15 Comprehensive Practice Questions with Step-by-Step Solutions

Q1. Divisibility by 72 (Missing Digits)

Question: If the 9-digit number 8354x26y2 is completely divisible by 72, find the value of $(3x - 2y)$ for the largest possible value of $y$.

Fast Method:
1. $72 = 8 \times 9$. Check divisibility by 8 on last 3 digits: 6y2.
2. For 6y2 to be divisible by 8: Since hundreds digit is 6 (even), last two digits y2 must be divisible by 8. Possible values of y2: $32, 72 \implies y = 3 \text{ or } y = 7$.
3. Largest value of $y = 7$.
4. Now test divisibility by 9 on 8354x2672:
  - Digit sum: $8 + 3 + 5 + 4 + x + 2 + 6 + 7 + 2 = 37 + x$.
  - Next multiple of 9 is $45 \implies x = 45 - 37 = 8$.
5. Value: $3(8) - 2(7) = 24 - 14 = \mathbf{10}$.
Answer: 10

Q2. LCM Model 1 (Common Remainder)

Question: What is the least number which when divided by 12, 16, 18, and 21 leaves a remainder of 7 in each case?

Fast Method:
1. Required number $N = \text{LCM}(12, 16, 18, 21) \times k + 7$.
2. Mental LCM:
  - $16 = 2^4$
  - $18 = 2 \times 3^2 \implies$ need $3^2 = 9$
  - $21 = 3 \times 7 \implies$ need 7
  - $12 = 2^2 \times 3 \implies$ already covered
  - $\text{LCM} = 16 \times 9 \times 7 = 144 \times 7 = 1008$.
3. Smallest number (for $k = 1$): $1008 + 7 = \mathbf{1015}$.
Answer: 1015

Q3. LCM Model 2 (Constant Difference)

Question: Find the least number which when divided by 20, 25, 35, and 40 leaves remainders 14, 19, 29, and 34 respectively.

Fast Method:
1. Check differences: $$20 - 14 = 6$$ $$25 - 19 = 6$$ $$35 - 29 = 6$$ $$40 - 34 = 6 \implies K = 6$$
2. Formula: $N = \text{LCM}(20, 25, 35, 40) - 6$.
3. Calculate LCM:
  - $40 = 2^3 \times 5$
  - $25 = 5^2 \implies$ need extra 5 $\implies 40 \times 5 = 200$
  - $35 = 5 \times 7 \implies$ need 7 $\implies 200 \times 7 = 1400$
  - $20$ is covered.
  - $\text{LCM} = 1400$.
4. $N = 1400 - 6 = \mathbf{1394}$.
Answer: 1394

Q4. HCF Difference Model (Unknown Remainder)

Question: Find the greatest number that will divide 1305, 4665, and 6905 leaving the same remainder in each case. Also, find that remainder.

Fast Method:
1. Take pairwise differences:
  - $4665 - 1305 = 3360$
  - $6905 - 4665 = 2240$
  - $6905 - 1305 = 5600$
2. Calculate $\text{HCF}(3360, 2240, 5600)$:
  - Difference between 3360 and 2240 is 1120.
  - Does 1120 divide 2240? $2240 = 1120 \times 2$ ✓
  - Does 1120 divide 3360? $3360 = 1120 \times 3$ ✓
  - Does 1120 divide 5600? $5600 = 1120 \times 5$ ✓
  - $\text{HCF} = \mathbf{1120}$.
3. Remainder: $1305 \div 1120 = 1$ with remainder $\mathbf{185}$.
Answer: Divisor = 1120, Remainder = 185

Q5. Real-Life LCM (Bells Ringing)

Question: Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, and 12 seconds respectively. In 30 minutes, how many times do they toll together?

Fast Method:
1. $\text{LCM}(2, 4, 6, 8, 10, 12) = 120$ seconds $= 2$ minutes.
2. Total duration $= 30$ minutes.
3. Number of times $= \lfloor 30 / 2 \rfloor + 1$ (including the initial toll at start!) $= 15 + 1 = \mathbf{16 \text{ times}}$. (⚠️ Common trap: Forgetting the starting toll at time $t = 0$. Always verify if the question says "excluding the start").
Answer: 16 times

Q6. Negative Remainder Shortcut

Question: What is the remainder when $(67^{67} + 67)$ is divided by 68?

Fast Method:
1. $67 \equiv -1 \pmod{68}$.
2. Expression becomes: $$(-1)^{67} + (-1) \pmod{68} = -1 - 1 = -2 \pmod{68}$$
3. Convert negative remainder to positive: $$68 - 2 = \mathbf{66}$$
Answer: 66

Q7. Fermat’s Little Theorem

Question: Find the remainder when $3^{102}$ is divided by 101.

Fast Method:
1. 101 is a prime number.
2. By Fermat's Theorem: $3^{100} \equiv 1 \pmod{101}$.
3. $3^{102} = 3^{100} \times 3^2 \equiv 1 \times 9 = \mathbf{9} \pmod{101}$.
Answer: 9

Q8. Remainder of Factorial Sum

Question: Find the remainder when $(1! + 2! + 3! + 4! + \dots + 100!)$ is divided by 12.

Fast Method:
1. Note that $4! = 24$, which is completely divisible by 12 ($24 \pmod{12} = 0$).
2. Every subsequent factorial $5!, 6!, \dots, 100!$ contains $4!$ as a factor, so all terms from $4!$ onwards have remainder 0!
3. We only need to sum terms before $4!$: $$1! + 2! + 3! = 1 + 2 + 6 = 9$$
4. Remainder $= 9 \pmod{12} = \mathbf{9}$.
Answer: 9

Q9. Units Digit with Borrowing Trap

Question: What is the unit digit of the expression: $7^{95} - 3^{58}$?

Fast Method:
1. For $7^{95}$:
  - Exponent $95 \pmod 4 = 3$.
  - Unit digit of $7^3 = 343 \implies 3$.
2. For $3^{58}$:
  - Exponent $58 \pmod 4 = 2$.
  - Unit digit of $3^2 = 9$.
3. Subtraction: $(...3) - (...9)$.
  - Borrow 1 from tens place: $13 - 9 = \mathbf{4}$.
Answer: 4

Q10. Trailing Zeros in Single Factorial

Question: Find the number of trailing zeros at the end of $200!$.

Fast Method:
1. Successively divide by 5: $$\lfloor 200 / 5 \rfloor = 40$$ $$\lfloor 40 / 5 \rfloor = 8$$ $$\lfloor 8 / 5 \rfloor = 1$$
2. Sum $= 40 + 8 + 1 = \mathbf{49}$.
Answer: 49

Q11. Highest Power of Composite Base in Factorial

Question: What is the highest power of 12 that divides $100!$?

Fast Method:
1. $12 = 2^2 \times 3^1$.
2. We need to check both powers of 2 and powers of 3:
  - For 3: $\lfloor 100/3 \rfloor + \lfloor 33/3 \rfloor + \lfloor 11/3 \rfloor + \lfloor 3/3 \rfloor = 33 + 11 + 3 + 1 = 48 \implies 3^{48}$.
  - For 2: $\lfloor 100/2 \rfloor + \lfloor 50/2 \rfloor + \lfloor 25/2 \rfloor + \lfloor 12/2 \rfloor + \lfloor 6/2 \rfloor + \lfloor 3/2 \rfloor + \lfloor 1/2 \rfloor$ $= 50 + 25 + 12 + 6 + 3 + 1 = 97 \implies 2^{97}$.
  - Number of $2^2$ available: $\lfloor 97 / 2 \rfloor = 48$.
3. Combining: $\min(48 \text{ from } 2^2, 48 \text{ from } 3) = \mathbf{48}$.
Answer: 48

Q12. Factors Breakdown (Total, Even, Odd, Sum)

Question: For the number 720: (a) How many total factors does it have? (b) How many even factors? (c) How many odd factors? (d) What is the sum of all its factors?

Fast Method:
1. Prime factorize 720: $$720 = 72 \times 10 = (2^3 \times 3^2) \times (2 \times 5) = 2^4 \times 3^2 \times 5^1$$
2. (a) Total factors: $$(4 + 1)(2 + 1)(1 + 1) = 5 \times 3 \times 2 = \mathbf{30}$$
3. (b) Odd factors: Ignore $2^4$, use only $3^2 \times 5^1$: $$(2 + 1)(1 + 1) = 3 \times 2 = \mathbf{6}$$
4. (c) Even factors: $$\text{Total} - \text{Odd} = 30 - 6 = \mathbf{24}$$ (Alternatively: $4 \times (2+1) \times (1+1) = 4 \times 3 \times 2 = 24$)
5. (d) Sum of factors: $$(2^0 + 2^1 + 2^2 + 2^3 + 2^4)(3^0 + 3^1 + 3^2)(5^0 + 5^1)$$ $$= (1 + 2 + 4 + 8 + 16)(1 + 3 + 9)(1 + 5)$$ $$= 31 \times 13 \times 6 = 31 \times 78 = \mathbf{2418}$$
Answer: (a) 30, (b) 24, (c) 6, (d) 2418

Q13. Divisibility by 11 with Alternating Digits

Question: If the 7-digit number 4a56b78 is divisible by 11 and $(a - b) = 3$, find the values of $a$ and $b$.

Fast Method:
1. Label positions:
  - Odd positions (1st, 3rd, 5th, 7th from right): $8, b, 5, 4 \implies \text{Sum}_1 = 17 + b$.
  - Even positions (2nd, 4th, 6th from right): $7, 6, a \implies \text{Sum}_2 = 13 + a$.
2. Difference: $(17 + b) - (13 + a) = 4 + (b - a)$.
3. Given $(a - b) = 3 \implies (b - a) = -3$.
4. Difference $= 4 + (-3) = 1 \ne 0 \text{ or multiple of } 11$.
5. This means difference must equate to $\pm 11$:
  - Case 1: $4 + (b - a) = 11 \implies b - a = 7 \implies a - b = -7$.
  - Case 2: $4 + (b - a) = -11 \implies b - a = -15$ (impossible).
  - With original equation $a - b = 3$: difference is $4 - 3 = 1$. For divisibility, $4 + b - a = 0 \implies a - b = 4$.
  - Since given $a - b = 3$, no integer digits satisfy both conditions. But if $(a - b) = 4$, then $4 - 4 = 0$ holds! E.g., $a = 7, b = 3$.
Answer: $a = 7, b = 3$ (when $a - b = 4$)

Q14. Wilson’s Theorem Application

Question: What is the remainder when $18!$ is divided by 19?

Fast Method:
1. 19 is a prime number.
2. By Wilson's Theorem: $(p - 1)! \equiv -1 \equiv (p - 1) \pmod p$.
3. Here $p = 19$, so $(19 - 1)! = 18! \equiv 18 \pmod{19}$.
Answer: 18

Q15. Minimum Number of Square Tiles (HCF Application)

Question: A courtyard 3.78 meters long and 5.25 meters broad is to be paved exactly with square tiles, all of the same size. What is the minimum number of tiles required?

Fast Method:
1. Convert dimensions to cm to avoid decimals: $$\text{Length} = 378 \text{ cm}, \quad \text{Breadth} = 525 \text{ cm}$$
2. To minimize the number of tiles, tile side must be the maximum possible $\implies \text{HCF}(378, 525)$.
  - $525 - 378 = 147$.
  - Factors of $147 = 3 \times 7^2$.
  - $378 / 21 = 18$ ✓, $525 / 21 = 25$ ✓.
  - Tile side $= \text{HCF} = \mathbf{21 \text{ cm}}$.
3. Minimum number of tiles: $$\text{Tiles} = \frac{\text{Total Area}}{\text{Area of 1 tile}} = \frac{378 \times 525}{21 \times 21} = 18 \times 25 = \mathbf{450}$$
Answer: 450 tiles

🔑 Exam Strategy & Trader Reflexes for Number System

Prioritize 10-Second Kills:
- Divisibility rules, unit digit cyclicity, and negative remainder problems take less than 15 seconds. Knock these down immediately to build bankable time for Data Interpretation.
Watch for the "+1" Trap in Repetitive Events:
- When bells toll or clocks tick, $\lfloor \text{Time} / \text{LCM} \rfloor + 1$ accounts for the starting event at $t=0$.
Compound Divisibility Protocol:
- When given $72$ or $88$, never test the sum of digits first. Always solve the last 3 digits for 8 first, as it narrows the unknown down to 1 or 2 options immediately.
Never Guess with Negative Marking:
- Futures First typically penalizes wrong answers with $-0.34$ to $-1$ mark. If a remainder problem involves an unfamiliar cycle and takes $>40$ seconds, use the "skip-fast" trader reflex and move to the next question.

01_number_system.md