Number Properties page 62 #14

[mcgarryseanm]

Q: If a and b are positive integers such that the remainder is 4 when a is divided by b, what is the smallest possible value of a+b?

My answer/logic: B must be at least 5. If it were 4, then it would be evenly divisible by 4, leaving no remainder. The smallest possible value for a is 9... 9/5 = 1 R4. Therefore the smallest possible value of a+b is 14.

However, the answer explanation states that the newer is 9...
"Since a/b has a remainder of 4, b must be at least 5. The smallest possible value for a is 4 (it could also be 9, 14, 19, etc.) Thus, the smallest possible value for a+b is 9."

A cannot POSSIBLY be 4. That would make a/b = 4/5 which does not leave any remainder whatsoever. What am I missing here?

[xerocoool]

hey,

Q: If a and b are positive integers such that the remainder is 4 when a is divided by b, what is the smallest possible value of a+b?

A : Dividend
B : Divisor
R : Remainder

All Integer values

Also, the remainder is always less than the divisor, in this case the range of remainder is 0 <= R < B (divisor)

Now since the remainder is 4 then our divisor has to be greater than 4 (following the above range). So the smallest possible value of our divisor in this case is 5

Now to find "A"

Number (Dividend "A") = Add the remainder (R) to the multiple of divisor (Divisor "B")

A cannot POSSIBLY be 4. That would make a/b = 4/5 which does not leave any remainder whatsoever. What am I missing here?

A = 4 + 5 (0) = 4 + 0 = 4 (Imp - 0 is a multiple of every positive integer, 0 divided by any non zero value gives us 0 remainder. And if we get a 0 remainder we know "A" is completely divisible by "B")

Now you can find range of values going by the above method

A = 4 + 5 (1) = 4 + 5 = 9 (9 divided by 5 gives remainder 4)
A = 4 + 5 (2) = 4 + 10 = 14 (14 divided by 5 gives remainder 4)

and so on.

But the smallest value of "A" in this case is 4, 4 divided by 5 gives us the remainder 4 and quotient of 0.

a+b = 4 + 5 = 9

Hope this helps

[tommywallach]

Hey Guys,

Great explanation from Xero. Just to be super clear, this is a common misconception. Actually, remainder is whatever's left over after you've counted how many times the denominator divides into the numerator. If the denominator is bigger than the numerator (so the fraction is "proper"), then the remainder is just WHATEVER the numerator is, regardless of what the denominator is:

2/3 --> remainder 2

2/5 --> remainder 2

2/13 --> remainder 2

etc.

Hope that helps!

-t