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81. Find the largest number which divides 245 and 1029 leaving remainder 5 in each case.

It is given that required number which divides 245 and 1029, the remainder is 5 in each case.

⇒ 245 - 5 = 240 and 1029 - 5 = 1024 are completely divisible by the required number.

Since, it is given that the required number is the largest number.

Therefore, it is the HCF of 240 and 1024.

Now, finding HCF by Euclid’s division algorithm.

Given integers are 240 and 1024.

Clearly 1024 > 240.

Therefore, it is the HCF of 240 and 1024 and 240, we get

II. Since, the remainder 64 ≠ 0, we apply division lemma to get

III. We consider the new divisor 64 and remainder 48 and apply division lemma to get

IV. We consider the new divisor 48 and new remainder 16 to get

V. The remainder at this step is zero. So, the divisor at this stage or the remainder at the previous stage i.e., 16 is the HCF of 245 and 1029.

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82. Find the largest number which divides 615 and 963 leaving remainder 6 in each case.

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83. What is the largest number that divides 626, 3127 and 15628 and leaves remainders of 1, 2 and 3 respectively.

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Long Answer Type

84. Three sets of Physics, Chemistry and Biology books have to be stacked in such a way that all the books are stored topic-wise and the height of each stack is the same. The number of Physics books is 105, the number of Chemistry books is 140 and the number of Biology books is 175. Assuming that the books are of the same thickness, determine the number of stacks of Physics, Chemistry and Biology books.

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