Agniveer CEE 2026 — Mathematics

Explanation: Step 1: Apply divisibility rule for 11: alternating sum of digits (from right to left) must be divisible by 11. Step 2: For 5742: (2 - 4 + 7 - 5) = 0, which is divisible by 11. Step 3: Verify: 5742 ÷ 11 = 522. Therefore, 5742 is divisible by 11.

Explanation: Step 1: √16 = 4. Step 2: √25 = 5. Step 3: √4 = 2. Step 4: (4 + 5) × 2 = 9 × 2 = 18. Therefore, the answer is 18.

Explanation: Step 1: Apply divisibility rule for 11: alternating sum of digits (from right to left) must be divisible by 11. Step 2: For 5742: (2 - 4 + 7 - 5) = 0, which is divisible by 11. Step 3: Verify: 5742 ÷ 11 = 522. Therefore, 5742 is divisible by 11.

Explanation: Step 1: A number divisible by both 4 and 6 must be divisible by their LCM. Step 2: Find LCM(4, 6): 4 = 2², 6 = 2 × 3, so LCM = 2² × 3 = 12. Step 3: Any number divisible by 12 is also divisible by 12 itself, and by all its factors (1, 2, 3, 4, 6, 12). Therefore, the answer is 12.

Explanation: Step 1: Apply divisibility rule for 8: a number is divisible by 8 if its last three digits form a number divisible by 8. Step 2: Check 3216: last three digits are 216. 216 ÷ 8 = 27 ✓ (divisible). Step 3: Check 4128: last three digits are 128. 128 ÷ 8 = 16 ✓ (divisible). Step 4: Check 5240: last thre…

Explanation: Step 1: Rewrite 125 as a power of 5: 125 = 5³. Step 2: So 5^(2x-1) = 5³, which means 2x - 1 = 3. Step 3: Solve for x: 2x = 4, therefore x = 2.

Explanation: Step 1: Apply divisibility rule for 3: sum of all digits must be divisible by 3. Step 2: Check 4827: 4 + 8 + 2 + 7 = 21, divisible by 3. Step 3: Check 5634: 5 + 6 + 3 + 4 = 18, divisible by 3. Step 4: Check 7249: 7 + 2 + 4 + 9 = 22, NOT divisible by 3. Step 5: Check 6381: 6 + 3 + 8 + 1 = 18, divisib…

Explanation: Step 1: Find LCM(6, 8): 6 = 2 × 3, 8 = 2³, so LCM = 2³ × 3 = 24. Step 2: Numbers divisible by both 6 and 8 are multiples of 24. Step 3: Find multiples of 24 between 100 and 200: 120, 144, 168, 192. Step 4: Count them: 4 numbers.

Explanation: Step 1: A number is divisible by 6 if it is divisible by both 2 and 3. Step 2: For 4,374: it is even (divisible by 2). Step 3: Check divisibility by 3: sum of digits = 4 + 3 + 7 + 4 = 18, which is divisible by 3. Step 4: So 4,374 is divisible by 6. Step 5: For 4,375: it is odd (not divisible by 2). …

Explanation: Step 1: Find the smallest number ≥ 100 divisible by 7: 100 ÷ 7 = 14.28..., so first number is 7 × 15 = 105. Step 2: Find the largest number ≤ 200 divisible by 7: 200 ÷ 7 = 28.57..., so last number is 7 × 28 = 196. Step 3: Count: numbers are 7 × 15, 7 × 16, ..., 7 × 28. Step 4: Count = 28 - 15 + 1 = …

Explanation: Step 1: A number divisible by both 4 and 6 must be divisible by their LCM. Step 2: LCM(4, 6) = 12 (since 4 = 2², 6 = 2 × 3, so LCM = 2² × 3 = 12). Step 3: Therefore, any number divisible by both 4 and 6 must be divisible by 12.

Explanation: Step 1: Apply divisibility rule for 8: a number is divisible by 8 if its last three digits form a number divisible by 8. Step 2: Check 4328: last three digits are 328. 328 ÷ 8 = 41, so 4328 is divisible by 8. Step 3: Check 5432: last three digits are 432. 432 ÷ 8 = 54, so 5432 is divisible by 8. Ste…

Explanation: Step 1: A number is divisible by 9 if and only if the sum of its digits is divisible by 9. Step 2: If a number leaves remainder 0 when divided by 9, it is divisible by 9. Step 3: Therefore, the sum of its digits must be divisible by 9.

Explanation: Step 1: ∛8 = ∛(2³) = 2. Step 2: ∜16 = ∜(2⁴) = 2. Step 3: ∛27 = ∛(3³) = 3. Step 4: Substitute: (2 × 2) / 3 = 4/3.

Explanation: Step 1: A number is divisible by 9 if and only if the sum of its digits is divisible by 9. Step 2: If a number is divisible by 9, it is also divisible by 3 (since 9 = 3²). Step 3: Therefore, the number must also be divisible by 3. The answer is: The number is divisible by 3.

Explanation: Step 1: Apply divisibility rule for 8: check if the last three digits form a number divisible by 8. Step 2: For 45,624: last three digits are 624. Step 3: 624 ÷ 8 = 78, so 624 is divisible by 8. Step 4: Therefore, 45,624 is divisible by 8.

Explanation: Step 1: A number divisible by both 4 and 9 must be divisible by their LCM. Step 2: Since 4 = 2² and 9 = 3², and gcd(4,9) = 1, we have LCM(4,9) = 4 × 9 = 36. Step 3: Therefore, any number divisible by both 4 and 9 must be divisible by 36. Step 4: Among the options, 36 is the only number that must div…

Explanation: Step 1: Divisibility rule for 9: a number is divisible by 9 if and only if the sum of its digits is divisible by 9. Step 2: If a number is divisible by 9, then the sum of its digits must be divisible by 9. Step 3: Since 9 = 3 × 3, any number divisible by 9 is also divisible by 3. Therefore, the sum …

Explanation: Step 1: The number must be divisible by both 8 and 5. Step 2: Find LCM(8, 5): Since 8 = 2³ and 5 = 5 (coprime), LCM = 8 × 5 = 40. Step 3: The smallest positive number divisible by both is 40.

Explanation: Step 1: Use the identity (a + b)² = a² + 2ab + b². Here a = √5 and b = √3. Step 2: a² = 5, b² = 3, and 2ab = 2 × √5 × √3 = 2√15. Step 3: Add: 5 + 3 + 2√15 = 8 + 2√15. Therefore, the answer is 8 + 2√15.