Every pair of positive integers a and b there exist a unique pair of whole numbers q and r such that a=bq+r give examples of a and b wherever possible satisfying.
a r=0
b q=0
c r>b
d If a<b what can be said about q and r.
Every pair of positive integers a and b there exist a unique pair of whole numbers q and r such that a=bq+r give examples of a and b wherever possible satisfying.
a r=0
b q=0
c r>b
d If a<b what can be said about q and r.
THEOREM : Given two integers a and b, with b ≠ 0, there exist unique integers q and r such that
a = bq + r and 0 ≤ r < |b| where |b| denotes the absolute value of b.
A) If r=0, a=bq only. So, q = a/b which implies b > 0 (by 0 ≤ r < |b).
So, for q to be a whole number, a is the multiple of b. That means every such pair will satisfy where a=nb, n is a whole number.
B) If q=0, a=r. So, b can be any possible integer. For possible value of a, 0 ≤ a=r < |b|
C) r > b is only possible when b is a negative integer and 0 ≤ r < |b| satisfies.
EXAMPLE - If a = 7 and b = −3, then q = −2 and r = 1, since 7 = −3 × (−2) + 1
D) If a<b, a<bq also when q ≠ 0 (because q is a whole number)
r is also whole number, so 'a' can never equal 'b+qr'
So, q=0, r=a
THEOREM : Given two integers a and b, with b ≠ 0, there exist unique integers q and r such that
a = bq + r and 0 ≤ r < |b| where |b| denotes the absolute value of b.
A) If r=0, a=bq only. So, q = a/b which implies b > 0 (by 0 ≤ r < |b).
So, for q to be a whole number, a is the multiple of b. That means every such pair will satisfy where a=nb, n is a whole number.
B) If q=0, a=r. So, b can be any possible integer. For possible value of a, 0 ≤ a=r < |b|
C) r > b is only possible when b is a negative integer and 0 ≤ r < |b| satisfies.
EXAMPLE - If a = 7 and b = −3, then q = −2 and r = 1, since 7 = −3 × (−2) + 1
D) If a<b, a<bq also when q ≠ 0 (because q is a whole number)
r is also whole number, so 'a' can never equal 'b+qr'
So, q=0, r=a
