Important Questions For Class 10 Maths Chapter 1 | Q4

Real Numbers | Largest Divisor that divides 2 numbers leaving 2 different remainders

This CBSE class 10 Maths Extra Practice Question is from Chapter 1 - Real Numbers. This is an importnat question and is a word problem that tests your understanding of divisors, remainders and idea of using HCF when a divisor divides 2 numbers leaving 2 different remainders.

Question 4 : What is the largest number that divides 967 and 1767 leaving remainders of 71 and 103 respectively?


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Explanatory Answer

Given: The two numbers are 967 and 1767.
Let x be the largest number that divides both 967 and 1767 and leaves remainders of 71 and 103 respectively.

Step 1 - Decoding the information given in the question

Remainder of \\frac{\text{967}}{\text{x}}) is 71.
So, 967 – 71 = 896 is divisible by x.
Remainder of \\frac{\text{1767}}{\text{x}}) is 103.
So, 1767 – 103 = 1664 is divisible by x.

So, x divides both 896 and 1664.
In other words, x is a factor of both 896 and 1664. i.e., x is a common factor of 896 and 1664.


Step 2 - Apply Euclid's Lemma to find the HCF

Step 1: Apply Euclid's Lemma on 1664 with 896 as divisor.
1664 = 896 × 1 + 768
The remainder is not zero.
Step 2: Apply Euclid's Lemma on 896 with 768 as divisor.
896 = 768 × 1 + 128
The remainder is not zero.
Step 3: Apply Euclid's Lemma on 768 with 128 as divisor.
768 = 128 × 6 + 0
The remainder is zero.
∴ The divisor of this step, 128 is the HCF of 896 and 1664.

128 is the largest number that divides 967 and 1767 leaving remainders 71 and 103 respectively.



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