W.W. Grainger (GWW) 10-Q 2006 Q1 Quarterly report

Exhibit 32(b)

CERTIFICATION PURSUANT TO

18 U.S.C. SECTION 1350,

AS ADOPTED PURSUANT TO

SECTION 906 OF THE SARBANES-OXLEY ACT OF 2002

I, P. O. Loux, Senior Vice President, Finance and Chief Financial Officer of W.W. Grainger, Inc. (“Grainger”), certify, pursuant to 18 U.S.C. Section 1350, as adopted pursuant to Section 906 of the Sarbanes-Oxley Act of 2002, that:

1. The Quarterly Report on Form 10-Q of Grainger for the quarter ended March 31, 2006, (the “Report”) fully complies with the requirements of Section 13(a) or 15(d) of the Securities Exchange Act of 1934; and

2. The information contained in the Report fairly presents, in all material respects, the financial condition and results of operations of Grainger.

/s/ P. O. Loux

P. O. Loux

Senior Vice President, Finance

and Chief Financial Officer

May 1, 2006