Answer :
To solve these problems, we need to find the greatest number (the greatest common divisor, GCD) that leaves a specific remainder when dividing given numbers. Here's a step-by-step explanation:
Find the greatest number which divides 19 and 25 leaving a remainder 1 in each case:
When a number leaves a remainder of 1, it means if we subtract 1 from each number, the original numbers become exact multiples of the divisor.
We transform the problem by considering the numbers 18 (19 - 1) and 24 (25 - 1).
Find the GCD of 18 and 24:
[tex]18 = 2 \times 3^2[/tex]
[tex]24 = 2^3 \times 3[/tex]The common factors are 2 and 3, so:
[tex]\text{GCD}(18, 24) = 2 \times 3 = 6[/tex]
Therefore, the greatest number is 6.
Find the greatest number which divides 245 and 1029 leaving a remainder of 5 in each case:
Transform the numbers to 240 (245 - 5) and 1024 (1029 - 5).
Find the GCD of 240 and 1024:
[tex]240 = 2^4 \times 3 \times 5[/tex]
[tex]1024 = 2^{10}[/tex]The common factor is 2, so:
[tex]\text{GCD}(240, 1024) = 2^4 = 16[/tex]
Therefore, the greatest number is 16.
Find the greatest number which divides 645 and 792 leaving a remainder 7 and 9 respectively:
Transform the numbers to 638 (645 - 7) and 783 (792 - 9).
Find the GCD of 638 and 783:
[tex]638 = 2 \times 11 \times 29[/tex]
[tex]783 = 3 \times 11 \times 29[/tex]The common factors are 11 and 29, so:
[tex]\text{GCD}(638, 783) = 11 \times 29 = 319[/tex]
Therefore, the greatest number is 319.
Find the greatest number which divides 35, 62 and 85 leaving remainders 8, 8 and 4 respectively:
Transform the numbers to 27 (35 - 8), 54 (62 - 8), and 81 (85 - 4).
Find the GCD of 27, 54, and 81:
[tex]27 = 3^3[/tex]
[tex]54 = 2 \times 3^3[/tex]
[tex]81 = 3^4[/tex]The common factor is 27 (since 27 is a factor of both 54 and 81), so:
[tex]\text{GCD}(27, 54, 81) = 27[/tex]
Therefore, the greatest number is 27.
Find the greatest number which divides 131, 160 and 223 leaving a remainder 7, 5 and 6 respectively:
Transform the numbers to 124 (131 - 7), 155 (160 - 5), and 217 (223 - 6).
Find the GCD of 124, 155, and 217:
[tex]124 = 2^2 \times 31[/tex]
[tex]155 = 5 \times 31[/tex]
[tex]217 = 7 \times 31[/tex]The common factor is 31, so:
[tex]\text{GCD}(124, 155, 217) = 31[/tex]
Therefore, the greatest number is 31.
By transforming the original numbers into numbers that divide perfectly by the unknown divisor, we can determine these greatest numbers efficiently.
