Introduction:
To find the largest number that will divide 398, 436, and 542, leaving remainders of 7, 11, and 15 respectively, we can use the concept of modular arithmetic and the greatest common divisor (GCD) of the given numbers.

Step 1: Finding the GCD:
The GCD of three or more numbers can be found by finding the GCD of the first two numbers, and then finding the GCD of the result with the next number, and so on.

Step 2: Applying Modular Arithmetic:
To find the largest number that divides the given numbers and leaves the specified remainders, we need to find a number that satisfies the following conditions:
1. Divisible by the GCD of the given numbers.
2. When divided by the GCD, leaves remainders of 7, 11, and 15 respectively.

Step 3: Applying the Chinese Remainder Theorem:
The Chinese Remainder Theorem (CRT) can be applied to find a solution that satisfies the given remainders. CRT states that if we have a system of congruences with pairwise relatively prime moduli, then there exists a unique solution modulo the product of the moduli.

Step 4: Solving the Congruences:
We can solve the congruences using the Extended Euclidean Algorithm or by inspection. For simplicity, let's solve the congruences by inspection.

398 ≡ 7 (mod x)
436 ≡ 11 (mod x)
542 ≡ 15 (mod x)

By inspection, we can see that x = 17 satisfies all three congruences.

Step 5: Finding the Largest Number:
Now, we have x = 17, which satisfies the given conditions. To find the largest number that divides 398, 436, and 542, leaving remainders of 7, 11, and 15 respectively, we need to find the GCD of the given numbers, which is 17.

Therefore, the largest number that will divide 398, 436, and 542, leaving remainders of 7, 11, and 15 respectively, is 17.

Conclusion:
The largest number that will divide 398, 436, and 542, leaving remainders of 7, 11, and 15 respectively, is 17. This number satisfies the conditions of being divisible by the GCD of the given numbers and leaving the specified remainders when divided by the GCD.

Hey mate here your answer

firstly we will subtract these no. as :-
398 - 7 = 391
436 - 11 = 425
542 - 15 = 527


so, we need 391, 425, and 527
BY USING EUCLID'S DIVISION ALGORITHM
425 = 391×1× 34
391 = 34 × 11 + 17
34 = 17 × 2 + 0
thus hcf = 17
hence the largest number is 17