A digital locksmith can only use prime numbers for an access code. Which of the following code sequences is valid?

Answer: 2, 3, 5. 1 is not prime. 9 is not prime (divisible by 3). 4 and 6 are composite. 2, 3, and 5 are all prime numbers.

An inspector must group 1,331 books equally into 11 boxes without leftovers. Can he do it?

Answer: Yes. Use the divisibility rule for 11: Sum of odd places (1+3=4) minus sum of even places (3+1=4) equals 0. Hence, 1331 is perfectly divisible by 11.

You plant trees in a huge circle numbered 1 to 100. If a storm destroys every tree standing on a multiple of 3, how many trees survive?

Answer: 67. Total trees = 100. Trees on multiples of 3 = 100 ÷ 3 = 33 (integer part). Trees destroyed = 33. Surviving trees = 100 - 33 = 67.

A baker notices that if he packs his cookies in bags of 2, 3, or 5, he always has exactly 1 cookie left. What is the minimum number of cookies he baked?

Answer: 31. The number must be LCM(2, 3, 5) + remainder. LCM of 2, 3, 5 is 30. Minimum cookies = 30 + 1 = 31.

During a drill, a commander asks soldiers to fall into a formation based on 'the smallest perfect square that is also a sum of two consecutive integers'. How many soldiers form the group?

Answer: 9. Check perfect squares: 4 (not sum of consecutive), 9 = 4+5 (sum of consecutive). So 9 soldiers.

You are setting a 6-digit password. To make it memorable, you decide the first 3 digits multiplied must equal the last 3 digits. If the password is 'abc024', what could 'a,b,c' be?

Answer: All of the above. The last 3 digits form 24. We need a × b × c = 24. 2×3×4=24. 1×4×6=24. 2×2×6=24. All work.

A neon sign flashes entirely based on the cyclicity of number 2. It flashes red on power 1, blue on power 2, green on power 3, and yellow on power 4. If this cycles infinitely, what color flashes on the 34th step?

Answer: Blue. Cyclicity is 4. Divide step 34 by 4. Remainder is 2. The 2nd color in the cycle is Blue.

At a fruit market, 3 apples and 2 bananas cost a certain total. If you write the calculation as 10 - 2 × 3 + 4 ÷ 2, what is the final numeric value of the cart?

Answer: 6. Apply BODMAS: First Division (4 ÷ 2 = 2), then Multiplication (2 × 3 = 6). Equation becomes: 10 - 6 + 2 = 6.

A spy writes an encrypted code where each digit is replaced by its 'face value' multiplied by the 'place value' of the 3rd column (hundreds). What happens to the digit 5 in the number 25,600?

Answer: It becomes 500. Face value of 5 is 5. Place value of 3rd column (hundreds) is 100. Encrypted value = 5 × 100 = 500.

You stack boxes in a pyramid: 1 on top, 2 in the next row, 3 in the next, up to a bottom row of 20 boxes. How many boxes do you have in total?

Answer: 210. Sum of first N natural numbers = N(N+1)/2. For N=20: 20(21)/2 = 10 × 21 = 210.

Find the unit digit of 24615 × 32110.

Answer: 6. Any power of a number ending in 6 ends in 6. Any power ending in 1 ends in 1. 6 × 1 = 6.

What is the difference between the place value and face value of 8 in the number 28,754?

Answer: 7992. Place value of 8 = 8000. Face value = 8. Difference = 8000 - 8 = 7992.

Simplify the expression: 100 ÷ 5 of 4 + 20.

Answer: 25. Solve 'Of' first: 5 × 4 = 20. Then divide: 100 ÷ 20 = 5. Then add: 5 + 20 = 25.

Which of the following numbers is completely divisible by 8?

Answer: 678312. Rule for 8: Check the last 3 digits. 162 is not div by 8. 218 is not. 312 ÷ 8 = 39 (Perfect division). So, C is correct.

If a number is divisible by 9 and 4, it MUST also be divisible by:

Answer: 36. If a number is divisible by two co-prime numbers (9 and 4), it is divisible by their product: 9 × 4 = 36.

Find the value of 1 - 2 + 3 - 4 + 5... up to 100 terms.

Answer: -50. Group in pairs: (1-2) + (3-4) + ... = (-1) + (-1) + ... There are 50 pairs. 50 × -1 = -50.

If 1A5A is divisible by 9, what is the value of the digit A?

Answer: 6. Rule of 9: Sum of digits = 1 + A + 5 + A = 6 + 2A. For 6 + 2A to be divisible by 9 (it could be 9, 18, 27). If 2A + 6 = 18 → 2A = 12 → A = 6.

A number leaves remainder 25 when divided by 48. What is the remainder when it is divided by 16?

Answer: 9. Since the new divisor (16) is a factor of the old divisor (48), just divide the old remainder (25) by 16. 25 ÷ 16 leaves remainder 9.

What is the unit digit of 5100 + 6100?

Answer: 1. 5 always ends in 5. 6 always ends in 6. Sum = 5 + 6 = 11. Unit digit is 1.

Which of the following is NOT a rational number?

Answer: π (Pi). π is an irrational number because it is non-terminating and non-repeating. 22/7 and 3.14 are rational approximations.

Find the sum of all odd numbers up to 100.

Answer: 2500. There are 50 odd numbers up to 100. Sum of first N odd numbers = N2. So, 502 = 2500.

How many prime factors exist in the expression 210 × 35 × 58?

Answer: 23. Total prime factors = Sum of the powers = 10 + 5 + 8 = 23.

Find the remainder when 38 × 39 × 40 is divided by 37.

Answer: 6. Use individual remainders: 38/37 rem 1. 39/37 rem 2. 40/37 rem 3. Multiply remainders: 1 × 2 × 3 = 6.

What is the unit digit of 1! + 2! + 3! + 4! + 5!?

Answer: 3. 1!=1, 2!=2, 3!=6, 4!=24, 5!=120. Sum digits: 1+2+6+4+0 = 13. Unit digit is 3.

How many zeroes are at the end of 50! (factorial)?

Answer: 12. Divide 50 by 5 = 10. Divide 10 by 5 = 2. Total zeros = 10 + 2 = 12.

Find the unit digit of 795 - 358.

Answer: 4. For 795: Remainder of 95/4 = 3 → 73 ends in 3. For 358: Remainder of 58/4 = 2 → 32 ends in 9. For subtraction 3 - 9, borrow 10 → 13 - 9 = 4.

A mysterious vault opens only when the code matches the remainder of 17200 divided by 18. What is the code?

Answer: 1. Using the Negative Remainder Theorem: 17 divided by 18 leaves a remainder of -1. So, (-1)200. Since 200 is an even power, the result is +1.

If n is an integer, the expression (n3 - n) is ALWAYS divisible by which largest number?

Answer: 6. Factorize: n(n2 - 1) = (n-1) × n × (n+1). This is the product of 3 consecutive integers. It must always be divisible by 3! = 6.

Simplify the cyclic nested fraction structure: 1 + 11 + 12.

Answer: 5/3. Solve bottom up: 1 + 1/2 = 3/2. The inverse is 2/3. Next layer: 1 + 2/3 = 5/3.

Find the HCF of (2100 - 1) and (2120 - 1).

Answer: 220 - 1. Algebraic Property: HCF of (am - 1) and (an - 1) is aHCF(m, n) - 1. So, HCF of 100 and 120 is 20. Result: 220 - 1.

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Numbers & Whole Operations

Master the building blocks of mathematics. Learn Divisibility Rules, Units Digit shortcuts, and BODMAS logic to eliminate calculation errors.

Expert Answer & Key Takeaways

Master the building blocks of mathematics. Learn Divisibility Rules, Units Digit shortcuts, and BODMAS logic to eliminate calculation errors.

1. Classification of Numbers

Every number belongs to a family. Real numbers split into Rational (fractions/integers) and Irrational (non-repeating decimals like π).

Key integer classifications:

Natural (N): 1, 2, 3...
Whole (W): 0, 1, 2... (Zero included).
Prime: Exactly 2 factors. 2 is the only even prime.
Composite: More than 2 factors. 1 is neither prime nor composite.

Example:

Q: Which of the following is the smallest odd prime number? (A) 1 (B) 2 (C) 3 (D) 9

Solution: 1 is not prime. 2 is prime but even. 3 is prime and odd. 9 is composite. Answer is 3.

2. The Sniper Method: Divisibility Rules

Check divisibility instantly without full division:

3 and 9: Sum of digits must be divisible by 3 or 9.
4 and 8: Last 2 digits (for 4) or 3 digits (for 8) must be divisible.
11: (Sum of odd-placed digits) - (Sum of even-placed digits) = 0 or multiple of 11.

Example:

Q: Is 9182736 divisible by 9?

Solution: Sum of digits: 9+1+8+2+7+3+6 = 36. Since 36 is divisible by 9, the entire number is divisible by 9.

3. Units Digit Cyclicity

The last digit of a power repeats in predictable cycles. Focus ONLY on the unit digit of the base.

0, 1, 5, 6: Always end in themselves (e.g., 5¹⁰⁰ ends in 5).
4, 9: Cycle of 2. (4^odd=4, 4^even=6).
2, 3, 7, 8: Cycle of 4. Divide the power by 4, take the remainder (R). If R=0, use power 4.

Example:

Q: What is the unit digit of 3⁴²?

Solution: Base ends in 3 (Cycle of 4). Divide power 42 by 4 → Remainder is 2. Calculate 3² = 9. The unit digit is 9.

4. The Core Model: VBODMAS

The strict order of operations prevents ambiguous answers.

V: Vinculum (Bar bracket over numbers; solve this first).
B: Brackets [ { ( ) } ] (Inner to Outer).
O: Of (Hidden Multiplication).
D/M: Division/Multiplication (Left to right priority).
A/S: Addition/Subtraction.

Example:

Q: Solve: 20 ÷ 5 of 2

Solution: 'Of' has priority over Division. First: 5 × 2 = 10. Then: 20 ÷ 10 = 2. (Many mistakenly do 20÷5=4, 4×2=8).

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Numbers & Whole Operations

Expert Answer & Key Takeaways

1. Classification of Numbers

2. The Sniper Method: Divisibility Rules

3. Units Digit Cyclicity

4. The Core Model: VBODMAS

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