Question

Find the largest number, smaller than the smallest four-digit number, which when divided by 4,5,6 and 7 leaves a remainder 2 in each case.

Answer 1

Answer:842

Step-by-step explanation:first of all multiply all the numbers that are 4,5,6,7

4*5*6*7=840

now as its a largest number which when divided lives remainder 0

so add 2 to the number that is 840+2=842.

Answer 2

Given : The number is largest number, smaller than the smallest four-digit number, which when divided by 4,5,6 and 7 leaves a remainder 2 in each case.

To find : The said number.

Solution :

The number is 842

We can simply solve this mathematical problem by using the following mathematical process.

Largest four digit number = 1000

So, the said number will be less than 1000

Now, let us take the LCM of the divisors.

4 = 2×2 = 2²

5 = 5¹

6 = 2×3 = 2¹×3¹

7 = 7¹

Factors with highest powers = 2²,3¹,5¹,7¹

So, LCM = (2² × 3¹ × 5¹ × 7¹) = (4×3×5×7) = 420

Now, the number 420 is divisible by each of the divisors, and it doesn't produce 2 as remainder. For having 2 as remainder, we have to add 2 with 420.

But, we cannot just add 2 with 420, and say that it is our required number. Because, the required number should be largest number, smaller than 1000. For solving this issue, we have to work with further multiples of 420 and add 2 with those further multiples. In this way, we have to find the largest number, smaller than 1000.

So, our required number will be :

= 420k + 2

(k can be any positive integer 1,2,3....)

• If k = 1 , the number = (420×1) + 2 = 422
• If k = 2 , the number = (420×2) + 2 = 842
• If k = 3 , the number = (420×3) + 2 = 1263

So, whenever the k becomes 3 the number exceeds the said limit. Which implies, the required number is produced when k becomes 2 (the number before 3).

So, the required number is = 842

Verification :

• 842 is less than 1000, and any exceeding value of k will produce a number greater than 1000. So, 842 is also the largest number possible.
• (842 ÷ 4) causes quotient = 210 , remainder = 2
• (842 ÷ 5) causes quotient = 168 , remainder = 2
• (842 ÷ 6) causes quotient = 140 , remainder = 2
• (842 ÷ 7) causes quotient = 120 , remainder = 2
• So, in each cases it produces a remainder of 2
• Which implies, the 842 is the said number.

Hence, the number is 842