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If p is a positive odd integer, what is the remainder when p [#permalink]

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23 Jul 2007, 07:25

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If p is a positive odd integer, what is the remainder when p is divided by 4?

(1) When p is divided by 8, the remainder is 5.

(2) p is the sum of the squares of two positive integers.

I see how (2) is sufficient, but I thought (1) was not sufficient because let p=3. Then p/4=0 R 3. p/8=0 R 5. So does the remainder thing only apply when you get an answet that is greater than 0?

If p is a positive odd integer, what is the remainder when p is divided by 4?

(1) When p is divided by 8, the remainder is 5.

(2) p is the sum of the squares of two positive integers.

I see how (2) is sufficient, but I thought (1) was not sufficient because let p=3. Then p/4=0 R 3. p/8=0 R 5. So does the remainder thing only apply when you get an answet that is greater than 0?

Yes, when a number (a) is divided by another number (b), and the remainder is >= 0, then a is >= b. So statement 1 states that when p/8, the remainder is 5. Therefore ,we know what p is greater than 8. So p can be 13, 21, ....

St1:
p = 8q1 + 5
q1 can be 1, then p = 13. p/4 -> R = 1
q1 can be 2, then p = 21. p/4 -> R = 1
q1 can be 3, then p = 29. p/4 -> R = 1
Seems R is always 1.
Sufficient.

St2:
p = x^2 + y^2
If x = 1, y = 2, then p = 5. p/4 -> R = 1
If x = 3, y = 2, then p = 13. p/4 -> R = 1
If x = 11, y = 20, then p = 521, p/4 -> R = 1
Seems R is always 1.
Sufficient.