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Question

For positive integers a and b, the remainder when a is divided by b is equal to the remainder when b is divided by a . Which of the following could be a value of ab ? I. 24 II. 30 III. 36

Answer Choices

A. $II only$
B. $III only$
C. $I and II only$
D. $II and III only$
E. $I, II, and III$

Steps

Explanation	Calculations	Help
$We translate "the remainder when a is divided by b is equal to the remainder when b is divided by a" into equal-remainder equations with integer quotients.$	$a = q b + r b = p a + r q, p are integers 0 \leq r r is less than b r is less than a$	Theory & Tactics Method Card TRANS5-F Click to view full details
$We compare the sizes of the two numbers to see when a common remainder can occur; we avoid the two unequal cases.$	$If a is less than b, then a = 0 \times b + a, so the remainder is a Also b = p a + r with 0 \leq r and r is less than a Equal remainders would force r = a, which is impossible since r is less than a If a is greater than b, then b = 0 \times a + b, so the remainder is b Also a = q b + r with 0 \leq r and r is less than b Equal remainders would force r = b, which is impossible since r is less than b So the only possibility is a = b, giving remainder 0$	Theory & Tactics Method Card TRANS5-F Click to view full details
$With the two numbers equal, the product is the same factor multiplied by itself. We now test the options.$	$ab = a \times a = a^{2}$
$Option I: check whether 24 can be written as the same integer times itself.$	$4^{2} = 16 5^{2} = 25 24 is between 16 and 25, so it is not of the form c \times c$
$Option II: check whether 30 can be written as the same integer times itself.$	$5^{2} = 25 6^{2} = 36 30 is between 25 and 36, so it is not of the form c \times c$
$Option III: check whether 36 can be written as the same integer times itself.$	$6^{2} = 36 Choose a = 6 and b = 6 to get ab = 36$

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Final Answer

III only

Question

For positive integers a and b, the remainder when a is divided by b is equal to the remainder when b is divided by a . Which of the following could be a value of ab ? I. 24 II. 30 III. 36

Answer Choices

A. $II only$
B. $III only$
C. $I and II only$
D. $II and III only$
E. $I, II, and III$

Steps

Explanation	Calculations	Help
$We translate "the remainder when a is divided by b is equal to the remainder when b is divided by a" into equal-remainder equations with integer quotients.$	$a = q b + r b = p a + r q, p are integers 0 \leq r r is less than b r is less than a$	Theory & Tactics Method Card TRANS5-F Click to view full details
$We compare the sizes of the two numbers to see when a common remainder can occur; we avoid the two unequal cases.$	$If a is less than b, then a = 0 \times b + a, so the remainder is a Also b = p a + r with 0 \leq r and r is less than a Equal remainders would force r = a, which is impossible since r is less than a If a is greater than b, then b = 0 \times a + b, so the remainder is b Also a = q b + r with 0 \leq r and r is less than b Equal remainders would force r = b, which is impossible since r is less than b So the only possibility is a = b, giving remainder 0$	Theory & Tactics Method Card TRANS5-F Click to view full details
$With the two numbers equal, the product is the same factor multiplied by itself. We now test the options.$	$ab = a \times a = a^{2}$
$Option I: check whether 24 can be written as the same integer times itself.$	$4^{2} = 16 5^{2} = 25 24 is between 16 and 25, so it is not of the form c \times c$
$Option II: check whether 30 can be written as the same integer times itself.$	$5^{2} = 25 6^{2} = 36 30 is between 25 and 36, so it is not of the form c \times c$
$Option III: check whether 36 can be written as the same integer times itself.$	$6^{2} = 36 Choose a = 6 and b = 6 to get ab = 36$

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Final Answer

III only