Number System

Product of two numbers = LCM * HCF. 26 * x = 182 * 13. x = (182 * 13) / 26 = 182 / 2 = 91.

Let the number be N = 221 * Q + 43. We want to find the remainder when N is divided by 17. Since 221 = 17 * 13, 221 * Q is divisible by 17. Thus, the remainder is the same as the remainder of 43 divided by 17. 43 = 17 * 2 + 9. The remainder is 9.

Find the HCF of (4061-2), (4185-3), and (4227-4), which is the HCF of 4059, 4182, and 4223. 4059 = 41 * 99, 4182 = 41 * 102, 4223 = 41 * 103. The HCF is 41.

The number must leave a remainder of 2 when divided by 3, 4, and 5. This means the number must be of the form LCM(3, 4, 5) * n + 2. LCM(3, 4, 5) = 60. Thus, the form is 60n + 2. Since 180n is a multiple of 60, 180n + 2 also satisfies the condition.

Sum of first n numbers = n(n+1)/2. For n=100, Sum = 100 * 101 / 2 = 5050. 5050 divided by 9: 5050 = 9 * 561 + 1. The remainder is 1.

Let the number be N. N = 4a + 3. a = 5b + 4. N = 4(5b + 4) + 3 = 20b + 16 + 3 = 20b + 19. Dividing by 20 leaves a remainder of 19.

55 is congruent to 1 mod 3. Therefore, 55^34 is congruent to 1^34 mod 3, which is 1.

Two-digit numbers leaving a remainder of 1 when divided by 4 follow the sequence 13, 17, 21, ..., 97. This is an arithmetic progression with first term a=13, last term l=97, and common difference d=4. The number of terms n is (97-13)/4 + 1 = 22. The sum is (n/2)(a+l) = (22/2)(13+97) = 11 * 110 = 1210.

The product of two numbers equals the product of their HCF and LCM. Therefore, 275 * x = 11 * 825. Solving for x gives x = (11 * 825) / 275 = 33.

The difference between the numbers is 34369 - 31513 = 2856. The divisor must divide 2856. Factors of 2856 include 294, 357, etc. Checking 3-digit factors, 294 is a candidate. 31513 / 294 gives remainder 97. 34369 / 294 gives remainder 97.

For any two numbers a and b, the product of the numbers equals the product of their HCF and LCM. Therefore, 55 * 99 = 11 * LCM. Dividing both sides by 11 gives 5 * 99 = 495.

Find the LCM of 12, 15, 20. LCM(12, 15, 20) = 60. The number is of the form 60k + 7. For k=1, the number is 67.