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Problem of the Week
Problem E and Solution
Number Triangle
Problem
The number 95 can be written 4 × 18 + 4 + 18 + 1. If a = 4 and b = 18, then we have written
95 in the form ab + a + b + 1, where a and b are positive integers and a ≤ b. Working with the
numbers 91 to 100 shown in the triangle, determine which of these integers can be written in
the form ab + a + b + 1 in the greatest number of ways, given that a and b are positive integers
and a ≤ b? Which integer, if any, in the triangle cannot be written in the form ab + a + b + 1,
given that a and b are positive integers and a ≤ b?
Solution
We can factor the expression by grouping pairs of terms and then common factoring as follows:
ab + a + b + 1
= a(b + 1) + (b + 1)
= (a + 1)(b + 1)
Then let N = ab + a + b + 1 = (a + 1)(b + 1) for integer values of N from 91 to 100. We are
basically asked to write N as the product of two factors in as many ways as possible so that
the first factor is ≤ the second factor. Then the value of a is one less than the value of the first
factor and the value of b is one less than the value of the second factor. We can ignore writing
the product where N is written as the product of 1 and itself since a + 1 represents the first
factor. If the first factor is 1, then a = 0. Since a ≥ 1, this is not allowed.
First we will look at the factors of the integers from 91 to 100.
N
91
92
93
94
95
96
97
98
99
100
Factors
1,7,13,91
1,2,4,23,46,92
1,3,31,93
1,2,47,94
1,5,19,95
1,2,3,4,6,8,12,16,24,32,48,96
1,97
1,2,7,14,49,98
1,3,9,11,33,99
1,2,4,5,10,20,25,50,100
Possible Products (excluding 1 × N )
7 × 13
2 × 46 = 4 × 23
3 × 31
2 × 47
5 × 19
2 × 48 = 3 × 32 = 4 × 24 = 6 × 16 = 8 × 12
none possible
2 × 49 = 7 × 14
3 × 33 = 9 × 11
2 × 50 = 4 × 25 = 5 × 20 = 10 × 10
Since 96 can be written as the product of two positive integer factors in the greatest number of
ways, it can be written in the form ab + a + b + 1 in the greatest number of ways. Since 97 can
be written as the product of two positive integer factors in the least number of ways, it can be
written in the form ab + a + b + 1 in the least number of ways. In fact, it cannot be written in
the form ab + a + b + 1 with 1 ≤ a ≤ b with both a and b integers. See a summary on the next
page.
The number 96 can be written in the form ab + a + b + 1, where a and b are positive integers
and a ≤ b, in 5 ways:
Since 96 = 2 × 48, a + 1 = 2 and b + 1 = 48.
It follows that a = 1, b = 47 and 96 = 1(47) + 1 + 47 + 1.
Since 96 = 3 × 32, a + 1 = 3 and b + 1 = 32.
It follows that a = 2, b = 31 and 96 = 2(31) + 2 + 31 + 1.
Since 96 = 4 × 24, a + 1 = 4 and b + 1 = 24.
It follows that a = 3, b = 23 and 96 = 3(23) + 3 + 23 + 1.
Since 96 = 6 × 16, a + 1 = 6 and b + 1 = 16.
It follows that a = 5, b = 15 and 96 = 5(15) + 5 + 15 + 1.
Since 96 = 8 × 12, a + 1 = 8 and b + 1 = 12.
It follows that a = 7, b = 11 and 96 = 7(11) + 7 + 11 + 1.
The number 97 cannot be written in the form ab + a + b + 1 with a and b are positive integers
and a ≤ b .
Since 97 is prime it can be written as the product of positive integer factors in only one way
with that the first factor less than or equal to the second factor. Then, 97 = 1 × 97, a + 1 = 1
and b + 1 = 97. It follows that a = 0. But a ≥ 1 so 97 cannot be written in the form
ab + a + b + 1 with a and b are positive integers and a ≤ b .
Each of the numbers 91, 93, 94 and 95 can only be written in the form ab + a + b + 1, where a
and b are positive integers and a ≤ b, in 1 way.
• 91 = 6(12) + 6 + 12 + 1
• 93 = 2(30) + 2 + 30 + 1
• 94 = 1(46) + 1 + 46 + 1
• 95 = 4(18) + 4 + 18 + 1
Each of the numbers 92, 98 and 99 can be written in the form ab + a + b + 1, where a and b are
positive integers and a ≤ b, in 2 ways.
• 92 = 1(45) + 1 + 45 + 1 = 3(22) + 3 + 22 + 1
• 98 = 1(48) + 1 + 48 + 1 = 6(13) + 6 + 13 + 1
• 99 = 2(32) + 2 + 32 + 1 = 8(10) + 8 + 10 + 1
The number 100 can be written in the form ab + a + b + 1, where a and b are positive integers
and a ≤ b, in 4 ways.
• 100 = 1(49) + 1 + 49 + 1 = 3(24) + 3 + 24 + 1 = 4(19) + 4 + 19 + 1 = 9(9) + 9 + 9 + 1