Entrance Exams
Govt. Exams
LCM(7, 11, 13) = 1001. Next multiple is 1001. 1001 - 1000 = 1. Actually 1000 ÷ 1001 remainder = 1000. Need 1001 - 1000 = 1. Recheck: 1000 mod 1001 = 1000, so add 1 gives 1001. Add 12 gives 1012 = 1001 + 11.
Let numbers be 3k and 5k where HCF(3k, 5k) = k = 4. Numbers are 12 and 20. Sum = 32.
Using Legendre's formula: ⌊27/3⌋ + ⌊27/9⌋ + ⌊27/27⌋ = 9 + 3 + 1 = 13.
Let numbers be a and b. (a+b)² - 4ab = (a-b)². So (a-b)² = 400 - 384 = 16, thus |a-b| = 4.
Digit sum = 9 + 9 + 9 + 9 = 36. Note: A number and its digit sum have the same remainder when divided by 9.
Numbers divisible by both 6 and 9 must be divisible by LCM(6,9) = 18. Between 1-100: 18, 36, 54, 72, 90. Count = 5.
Let numbers be x, x+2, x+4, x+6, x+8. Sum = 5x + 20 = 120, so x = 20. Largest = 20 + 8 = 28.
Since 9 and 11 are coprime (HCF = 1), the number must be divisible by 9 × 11 = 99.
72 = 2³ × 3². Number of divisors = (3+1)(2+1) = 4 × 3 = 12.
61 is only divisible by 1 and itself. 51 = 3 × 17, 57 = 3 × 19, 63 = 9 × 7 are composite numbers.