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Chapter 1: Number Systems, Divisibility & Remainder TheoremsCore 12 mins

Lesson 1: Divisibility by 72, 88 & 99 (Co-Prime Factoring)

Solve 8-digit and 9-digit numbers divisible by 72 or 88 by splitting into co-prime pairs (8 & 9, 8 & 11).

Official Commission Weightage Radar•1990–2026 Shift Matrix

High Yield HotspotCore Tier

SSC CGL Tier 1/2TCS iON

2–3 Qs (4–6 Marks)

Tier 2: 3–4 Qs (9–12 Marks)

RRB NTPC & ALPRailways

2–3 Questions

CBT 1 & 2 Direct Questions

Recurrence92%

92% of Shifts

Repeated in almost every slot

Target Pace25–35s

25–35s

Topper Hall Budget Limit

Core Formulas, Axioms & Memory Shortcuts1-Click Copy

Essential equations tested in TCS iON Commission shifts

2 Formulas

Divisibility by 8

If Hundreds is even: TU div by 8 | If Hundreds is odd: (TU + 4) div by 8

TU = last 2 digits

Divisibility by 9

Digital Sum of all digits = 9k

Cast out 9s

First Principles: Co-Prime Factorization Rule

To test divisibility by any composite number N = a * b, a and b MUST be co-prime (i.e., GCD(a, b) = 1). • Divisibility by 72 = Check 8 and 9 (since GCD(8, 9) = 1). Checking 6 and 12 fails because numbers like 12 are divisible by both but NOT by 72. • Divisibility by 88 = Check 8 and 11 (GCD(8, 11) = 1). • Divisibility by 99 = Check 9 and 11 (GCD(9, 11) = 1).

Exam Pro Tip:Always factorize into relatively prime pairs; never use overlapping common factors.

The 1-Second Hundreds-Digit Rule for Divisibility by 8

A number is divisible by 8 if its last 3 digits (...H T U) form a multiple of 8. AIR-1 Shortcut: 1. Look at H (the hundreds digit): - If H is EVEN (0, 2, 4, 6, 8): Just check the last two digits (TU). If TU is divisible by 8, the entire number is. - If H is ODD (1, 3, 5, 7, 9): Add 4 to TU. If (TU + 4) is divisible by 8, the entire number is! Example: In 48792, H = 7 (odd). Test: 92 + 4 = 96. Since 96 is 8 * 12, 48792 is divisible by 8 instantly without dividing 792!

Exam Pro Tip:If hundreds digit is odd, add 4 to the last two digits. Eliminates 3-digit long division entirely.

Divisibility by 11: Alternating Sum & Sign Parity

A number is divisible by 11 if the difference between the sum of digits at odd places and even places is either 0 or a multiple of 11. Formula: |(Sum of Odd-place digits) - (Sum of Even-place digits)| = 11k (k in {0, 1, 2, ...}) Topper Technique: Write digits in pairs from right to left. Add each 2-digit block. If sum is divisible by 11, the number is.

Exam Pro Tip:Remember that the difference can be negative in raw subtraction; take modulus or add 11.

AIR-1 Digital Root (Casting out 9s) Speed Protocol

For divisibility by 9, sum of digits must equal 9k. Never sum numbers sequentially (e.g. 7+8=15...). Cast out all digits summing to 9 instantly: Ignore all 9s, 8+1, 7+2, 6+3, 5+4 pairs. The remaining sum gives the digital root in under 3 seconds.

Exam Pro Tip:Casting out 9s reduces a 10-digit number to a single-digit modulo in 3 seconds flat.

AIR-1 Exam Shift Solved Masterclass3 Shifts

Real commission shift problems solved using both standard textbook and 15-second topper methods.

SSC CGL 2024 Tier 1 (12 Sept Shift 2)Foundation

⚡ +90s Saved

Target: ~15s

Official Exam Question

If the 8-digit number 789x531y is completely divisible by 72, find the value of (5x - 3y) for the largest possible value of y.

🎯Option Elimination Radar (5-Sec Bypass)

Bypass 2–3 options before solving

1. Unit Digit / Tens Digit Filter:

Multiply or sum only the last digits. If options end with distinct digits (e.g. 2, 4, 6, 8), zero full calculation needed.

2. Divisibility & Digital Root:

Check factors of 3, 7, 9, 11 (e.g. Area with π = 22/7 must divide by 11). Eliminates non-multiples in 3 seconds.

3. Extreme Value & Range Check:

Round numbers to nearest 10 or 100. If 49 × 31 ≈ 1500, drop options below 1400 or above 1600 instantly.

4. Factor & Ratio Alignment:

In ratio, ages, and mixture questions, the final answer must divide by the target ratio component.

⚡AIR-1 Shortcut Method: Odd Hundreds + Digital Root Casting

⏱ 15s solve

Hundreds digit 3 is odd → 1y + 4 is multiple of 8 → 12+4=16 → y=2 in 3 sec. Cast out 9s: 9 gone, 7+2=9 gone, 8+1=9 gone → remaining 5+3+x = 8+x → x=1. 5(1)-3(2) = -1. Total time: 12 seconds.

Examiner Trap & Common Distractor

Students assume y can take multiple values like in other shifts and waste time searching for y=6, failing to notice 316 is not divisible by 8.

Factor 72 into Co-Primes

72 = 8 × 9. Both conditions must be satisfied independently.

Find Largest y using Divisibility by 8

Last 3 digits are 31y. Hundreds digit is 3 (ODD). By the odd hundreds rule: (1y + 4) must be divisible by 8. Possible values of 1y are 12 (12+4=16, div by 8) or 16 (16+4=20, no) or 19. If y=2: 12+4=16 (valid). If y=6: 16+4=20 (invalid). If y=0 to 9, testing 31y: 312/8 = 39 (valid, y=2). Next is 312 + 8 = 320 (different hundreds), 312 - 80... Testing y in 31y: 312 is div by 8 (y=2). What about y=6? 316/8 = 39.5. Hence, the only single digit for which 31y is divisible by 8 is y = 2.

Find x using Divisibility by 9

Sum of digits: 7 + 8 + 9 + x + 5 + 3 + 1 + 2 = 35 + x. For (35 + x) to be divisible by 9, x must be 1 (35 + 1 = 36).

Calculate Target Expression

5x - 3y = 5(1) - 3(2) = 5 - 6 = -1.

Interactive Visual Concept BlueprintVisual Proof

Exam blueprints & geometric proportions tailored to: Lesson 1: Divisibility by 72, 88 & 99 (Co-Prime Factoring)

Unit Digit Cyclicity Clock & Co-Prime Factoring:Mod 4 Power Cycle

Cycle of 1 (Identical)

0, 1, 5, 6

Always retain unit digit regardless of power (e.g. 5ⁿ = 5, 6ⁿ = 6)

Cycle of 2 (Odd/Even)

4, 9

4¹=4, 4²=6 (even:6, odd:4) | 9¹=9, 9²=1 (even:1, odd:9)

Cycle of 4 (Mod 4)

2, 3, 7, 8

Divide power by 4. Use remainder as exponent: Power mod 4

Alternative Problem-Solving Approaches3 METHODS COMPARED

Textbook Proof vs 15-Sec Vedic Shortcut vs 5-Sec Option Elimination tailored to: Lesson 1: Divisibility by 72, 88 & 99 (Co-Prime Factoring)

120s (Book)→15s (Vedic)→5s (Hack)

Representative Commission Exam Question:

If the 8-digit number 789x531y is completely divisible by 72, find the value of (5x - 3y) for the largest possible value of y.

Odd Hundreds + Digital Root Casting⚡ Topper Shortcut

15 Seconds

Direct formula and ratio application bypassing scratchpad calculations.

Exam Hall Caveat: Must memorize the core pattern condition; verify problem matches the standard shortcut formula.

Actual Shift PYQs on this LessonOfficial Commission Keys

Test your mastery on real questions asked by TCS in recent shifts

Explore 37-Yr Bank

SSC CGL 2024 Tier 1 • Shift 2 (12 Sept 2024)

Find the value of x if the 9-digit number 83P93678Q is completely divisible by 72, given that P and Q are positive single digits and P > Q.

P = 5, Q = 4

P = 6, Q = 4

P = 7, Q = 2

P = 8, Q = 0

Step-by-Step Commission Method:

Step 1: 72 = 8 x 9 (coprime factors). Step 2: Divisibility by 8 requires last 3 digits '78Q' divisible by 8. 784 / 8 = 98 (Q = 4) or 782/8 = 97.75... 784 gives Q = 4. Step 3: Sum of digits: 8+3+P+9+3+6+7+8+4 = 48 + P divisible by 9 -> 48 + 6 = 54 (P = 6). Since condition requires P > Q (6 > 4), valid solution is P = 6, Q = 4.

⚡ AIR-1 5-Second Topper Shorthand:

Exam Trick: 78Q div 8 -> check Q = 4 (784/8 = 98). Digital root of other digits sum to 3 -> P + 4 + 3 = 9k -> P = 2 or 6. Took 8 seconds.

RRB NTPC CBT 2 • Pay Level 6 Official Shift

What is the remainder when (7^84 + 5) is divided by 342?

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